Operator Methods and SU ( 1 , 1 ) Symmetry in the Theory of Jacobi and of Ultraspherical Polynomials

Starting from general Jacobi polynomials ( ) ( ) , n P u α β we derive for the Ultraspherical polynomials ( ) ( ) , n P u α α as their special case a set of related polynomials ( ) n G x α which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.


Introduction
The applications of the Classical Orthogonal polynomials in physical sciences are immense.To the Classical Orthogonal polynomials belong in one-dimensional case (one variable) the Jacobi polynomials with their special case of Ultraspherical polynomials the last including the Legendre polynomials and the Chebyshev polynomials of first and second kind and on the other side the Hermite and Laguerre polynomials.The last are better classified as two-dimensional polynomials with relicts and close transformation relations to both the Hermite and the Laguerre polynomials in one-dimensional case.
One of the best older sources about Classical Orthogonal polynomials in general theory and to the Jacobi polynomials, in particular, is the monograph of Szegö [1] who also gave the modern and most general definition of this class of polynomials with the symmetries already implemented in the notation ( ) ( ) and with the best standardization.A recommendable older representation with a chapter about Jacobi polynomials which uses already its modern definition is contained in the book of Jackson [2].A very comprehensive work with a chapter about the Classical Orthogonal polynomials is in the second volume of the monographs of Bateman and Erd'elyi [3] about Higher Transcendental Functions and also very comprehensive are the Collections of Formulae and Tables of Magnus, Oberhettinger and Soni [4] and of Gradshteyn and Ryzhik [5] and, furthermore, the collaborative work of Abramomitz and Stegun [6] with the chapter of Hochstrasser about Orthogonal polynomials.A monograph from Kusnetsov [7] has a long chapter about Orthogonal polynomials but it is not translated into English.However, the next two monographs to cite are translated into English.A very readable book with a large chapter about Orthogonal polynomials which includes many of their applications is that of Lebedev [8] and also another book in Russian of this time from Nikiforov and Uvarov [9] presents the general theoy of the Classical Orthogonal polynomials and its specialization together with applications in an appealing way.Also very readable is the monograph of Rainville [10] about special functions including the general theory of orthogonal poynomials and more detailed representations about the Jacobi polynomials and of ultraspherical polynomials or equivalently the Gegenbauer polynomials and their special cases of Legendre polynomials.In the work of Luke [11] we also find a large representative chapter about orthogonal polynomials with the general theory and with detailed consideration of Jacobi polynomials and of their special cases.A derivation of all one-dimensional Classical Orthogonal polynomials from a general definition and with proof of their completeness one finds in the monograph of Chihara [12].A representation of the Classical Orthogonal polynomials including the Jacobi polynomials (with notation ( ) formulae difficult to find at other places is the monograph of Srivastava and Manocha [15].
In more recent time, the collaborative NIST-handbook [16] of Mathematical Functions with the chapter of Koornwinder, Wong, Koekoek and Swarttouw [17] about Orthogonal polynomials is a very comprehensive work.A modern introduction to Orthogonal polynomials is contained in the monograph of Andrews, Askey and Roy [18] (see also [19]).
The representation of the Jacobi polynomials in a chapter of the monograph of Carlson [20] is a little uncommon and unusual but possibly more general as in other representations (biorthogonality with Jacobi functions) and the author generally does not write the upper indices of the Jacobi polynomials (it is difficult then to consider contiguous relations where they change in one formula) and it deals with also Laguerre polynomials under the heading of the chapter about Jacobi polynomials.A special monograph about the Chebyshev polynomials with their application in mathematical approximation theory but with only very shortly mentioning (p.35) their superclasses of Ultraspherical polynomials, Gegenbauer polynomials and of Jacobi polynomials is that of Rivlin [21].In short form one finds a lot of formulae for Orthogonal polynomials also in a book of Bell [22] which we cite mainly for a formula to Hermite polynomials (see footnote in Section 5).Possibly, there exist further excellent representations of the Orthogonal polynomials.
Many classes of polynomials and of Higher Transcendental Functions possess a group-theoretical background and were intensively studied in application of quantum mechanics to the theory of angular momentum that means to the irreducible representations of the three-dimensional rotation group ( )  (Spherical harmonics), in particular, by Wigner [23], Weyl [24] and Van der Waerden [25] as pioneers.The classical representation from the mathematical side is given in the monograph of Vilenkin [26].
It rests to a considerable part on the work of the Russian school of mathematicians to the representations of groups, in particular, of Gelfand, Naimark and others and of Vilenkin himself (e.g., [27] [28]).Other representations of a group-theoretical background of Orthogonal polynomials and Special functions are given, e.g., by Miller [29] [30] [31] and by Van der Jeugt [32] in the collection [33].The monograph of Dunkl [34] is about Orthogonal polynomials of several variables (see also already cited [14]) but it treats in addition their grouptheoretical background.
The main purpose of this article is to join an approach to the Ultraspherical polynomials which leads, in particular, to an alternative definition of this class of Furthermore, we find an operator identity for the more general case of Jacobi A. Wünsche polynomials which among others leads to a convolution identity for the Jacobi polynomials similarly to the Vandermond convolution formula for the binomial coefficients.We also have accumulated in the past some more or less new applications of Ultraspherical polynomials mainly connected with 2D-matrices and functions of them and of their special cases of Chebyshev polynomials which, however, we cannot include into this article.In recent time we elaborated in detail the Weyl correspondence of classical phase-space functions to quantum-mechanical operators where the Jacobi polynomials find their application in many facets and not only in their simplest special cases.A main result was shortly communicated in [35] but the material for representation became too much to be packed into present work and we intend to make this separately.
Some results and formulae in the article are formulated for the whole set of Jacobi polynomials and some only for its subset of Ultraspherical polynomials.
The reason for this restriction to Ultraspherical polynomials was in these cases that we could not obtain up to now a generalization for the whole set of Jacobi polynomials which may be fairly difficult as some generating functions for both sets suggest.

Basic Relations for General Jacobi Polynomials
We give in this Section without a proof a few basic known formulae for the general Jacobi polynomials ( ) ( ) . The integers ( ) , 0,1, 2, n n =  denote the degree of the polynomials in the variable u which is a real or complex variable and α and β are two parameters which usually take on real values.
A lot of formulae for them can be found in [1] and in short form in [3] [5] [6] [17].The Ultraspherical polynomials ( ) ( ) The Jacobi polynomials were defined in the now generally accepted notation ( ) ( ) by Szegö [1] in the Rodrigues-like form for Classical Orthogonal polynomials (e.g., [3]) as follows from which by applying the Leibniz rule for the multiple differentiation of a product of functions follows immediately the explicit representation and if we bring the function ( ) The standardization of the Jacobi polynomials is In [1] are also given relations to older definitions by Jacobi, Jordan, Stieltjes, Using two free parameters γ and δ one may insert an intermediate step in the transition from definition (2.1) to representation (2.3) according to This relation becomes important in Section 12 when we derive a convolution identity for Jacobi polynomials.
The Jacobi polynomials ( ) ( ) where starting from the basic definition of the Hypergeometric series of , ; ; F a b c z one has to apply the transformation of the factorials (e.g., [36], chap. (2.1.4),Equation ( 22)) and which goes back to Gauss the Jacobi polynomials can be explicitly represented also by or equivalently by substitution j n k = − of the summation index The transition from (2.7) to (2.8) can also be made directly on the level of the polynomials that is demonstrated in Appendix A.
From (2.1) follows the symmetry property .
In connection of (2.7) and (2.8) with this symmetry one may write down 4 equivalent definitions of the Jacobi polynomials by the Hypergeometric function , ; ; F a b c z which all are given in [3] (Chap.10.8, Equation ( 16)).
By differentiation of ( ) ( ) with respect to variable u one finds (most easily) using ( From this follows the expansion of ( ) ( ) Apart from the special case α β = which we consider beginning with Section 3 and a few other special cases which we do not discuss here it is not possible to give for the Jacobi polynomials ( ) ( ) of argument zero simple formulae of multiplicative type others than the sum formulae which follow from the basic explicit relations for ( ) ( ) by specialization.
We abbreviate the weight function of the Jacobi polynomials ( ) ( ) Using the recurrence relation which can be checked by the given explicit representations for the Jacobi polynomials one may derive the following well-known second-order differential equation for the Jacobi polynomials (e.g, [1] We introduce now the functions ( ) ( ) They are orthonormalized in the interval [ ] Connected with these orthonormality relations are the completeness relations in the interval [ ] In both relations (2.17Using transformation relations for the operators in the differential equation one may transform it, for example, to the following form with a self-adjoint operator in front of ( ) ( ) . A slightly modified in form but fully equivalent representation of this differential equation is derived by Szegö [1] (Chap.IV, Equations (4.24.1) and (4.24.2)).
We mention here shortly that Equation (2. 19) is equivalent to the Schrödinger equation for the Pöschl-Teller potentials (e.g., [37] [38]).For this purpose we make in (2.19) the substitution From the orthonormalization of the functions ( ) ( ) follows that now the states ( ) ( )  We mention that in [37] and also in [38] the eigensolutions are orthonormalized in the interval 0 π x ≤ ≤ according to .
The first derivative of ( ) ( ) , 2 which explicitly possesses the form  x = are supposed to be impenetrable according to boundary conditions in (2.21) and in other case the results have to be modified in dependence on the boundary conditions.Furthermore, there are peculiarities with the infinite walls at the boundaries which become infinite abysses at 0 x = for 2 1  4   α < and at π x = for 2 1 4 β < .To study all this in systematic way and, for example, also the possible continuation of the potentials for all real x in periodic way is not our intention here.
The completeness relation (2.18) allows to make expansions of functions in series of the Jacobi polynomials ( ) ( ) or, more generally, in series of the set of functions with additional widely arbitrary parameter λ according to ( ) f u is well-behaved enough to guarantee the existence of the involved integrals.
As example for the inversion of the expression of the Jacobi polynomials ( ) ( ) we found (see also [10], p. 62, Equation ( 2)) where ( ) , α β are parameters which can be chosen arbitrarily.The expansion of n u in a sum over Jacobi polynomials is Apparently, the last relation is new.
From a general transformation relation of the Hypergeometric function , ; ; F a b c z where the variable z is transformed to the variable , first two of Equations (9.131), [36], Equation 2.1.( 23) and ( 24)) follows via the connection with the Jacobi polynomials the following general transformation with change of the variable u of the Jacobi polynomials to and with symmetry (2.10) (Szegö [1]) with fixed points 1, 3 u = + − .The argument u is here transformed by a certain fractional linear transformation (Möbius transformation) to the new argument.
There exists yet another transformation relation for the Jacobi polynomials ( ) ( ) which is only possible if one of the upper indices α or β is an integer k n ≥ − and which we write in the form Its origin comes also from one of the general transformation relations for the Hypergeometric function , ; ; F a b c z if in addition to a or b equal to negative integers n − (polynomial case) the third parameter c is equal to an integer k n ≥ − (see, e.g., [5], third of Equations 9.131, [36], Equation 2.1.(23)).
In other case the relation is also true but leads outside from the Jacobi polynomials to the corresponding Hypergeometric function since the lower index n k + is then no more an integer.
The basic recurrence relation for the Jacobi polynomials with fixed upper indices possesses the form (e.g., [1] [3] and others) and the corresponding relation for raising the index There is a large number of contiguous relations where in addition to the lower index n or alone also one or both upper indices ( ) , α β are changed by steps of 1.We are mainly concerned in this paper with relations where the upper indices ( ) , α β are fixed parameters and therefore we do not give them.

Basic Relations for Ultraspherical Polynomials
As already said the special case α β = of the Jacobi polynomials ( ) ( ) In the following we do not translate every formula for the Ultraspherical poly- The Gegenbauer polynomial In this and many other formulae in literature it is often used the Pochhammer symbol as abbreviation for the so-called "shifted" factorials or Gamma function.
From (2.10) follows for the symmetry of the Ultraspherical polynomials This means that ( ) ( )  are even poly- nomials of u which therefore depend only on 2 u .This also means that Concerning the transformation relations following from the transformation relations of the Hypergeometric function , ; ; F a b c z the case α β = possesses a specifics with no correspondence in the general Jacobi polynomials ≠ .This is one of the cases where the application of the so-called quadratic transformation relations of Gauss and Kummer is possible.
Specifically, this is the transformation relation given in [36] (2.1.5,Equation ( 28 where the argument u is transformed in a quadratic function of u as new argument.This shows also that the Jacobi polynomials with unequal upper the upper indices must not necessarily be equal for representations of Ultraspherical polynomials.
The weight function for Ultraspherical polynomials is From (2.15) we obtain the differential equation for Ultraspherical polynomials ( ) One of the forms of the differential equation for Ultraspherical polynomials multiplied with the square root of the weight function is The orthonormalized functions ( ) ( ) defined by (see (2.16)) satisfy the same differential equation as in (3.9) that means The orthonormality relation for the Ultraspherical polynomials possesses the form , and the completeness relation for the interval [ ] The operation of lowering the degree of the Ultraspherical polynomials , and the corresponding operation of raising the degree specialized from (2.34) They form the starting point in Section 9 for the explicit determination of the connection of the Ultraspherical polynomials to the ( )

Explicit Expressions for the Ultraspherical Polynomials
We now derive and compile explicit sum representations for the Ultraspherical polynomials.Using the symmetry (3.3) we find from (2.9) The factor ( ) A. Wünsche Therefore after changing the order of summations we find that after evaluation of the inner sum leads to the known explicit representation usually found in other indirect way.A direct proof of this evaluation is given in Appendix B. 3 Using the duplication formula for the argument of the Gamma function one finds from (4.4) the equivalent representation Thus we obtained the two equivalent basic representations of the Ultraspherical polynomials ( ) ( ) by an expansion in powers of the variable u that was not possible in such simple way for the general Jacobi polynomials ( ) ( ) From (4.4) follows for the argument 0 u = of ( ) ( ) where the vanishing for odd n is due to the symmetry (3.4).If we write down the expansion (2.12) in special case α β = we have after changing the index of and by comparison of (4.4) with (5.2) or directly from (4.6) we find The expression for ( ) (or its generalization for non-integer α ).It is interesting that we met already the case 0 α = when considering the expansion of ( ) ( ) ( ) x iy x iy + − [40] (and papers cited therein).

Alternative Representation of Ultraspherical Polynomials
We derive in this Section an alternative explicit representation of the Ultraspherical polynomials which becomes important for our further considerations.
We start from the representation (4.4) of ( ) ( ) which we write in a slightly different form and make the following transformations using the binomial formula and reordering of the arising double sum With the evaluation of the inner sum which we make in detail in Appendix B we obtain the following substantial new representation of the Ultraspherical polynomials One can also make the inverse transition from this representation to the representation (4.4) by expansion of ( ) into a Taylor series in powers of 2 u and evaluation of the arising (inner) sum.
We make now the following coordinate transformation u The variable u in the interval [ ] 1, 1 − + is here extended to the whole real axis ( ) , −∞ +∞ for the variable x .If we make this substitution of the variable u in (2) according to (3) we find the following relation is a polynomial of the variable x of degree n and we introduce for it multiplied by a factor for practical purpose a new notation or in representation by the Hypergeometric function , ; ; In Appendix C we give tables of the polynomials Formula (5.4) can be represented in the form We have now obtained a very interesting formula for the polynomials They can be obtained by application of the (integral) operator (5.9) onto the monomials n x .Although this operator depends on α as a fixed parameter it does not depend on the degree n of the polynomial 4 .4   This is in great analogy to the Hermite polynomials ( ) n H x which can be generated from the monomials x n by the formula [35] [40] ( ) ( )  as an alternative definition and which in some of our paper of the last years plaid an important role.The oldest source for this formula which we found up to now, however, without a citation there is the book of Bell [22] (p.159).
From the differential Equation (3.8) for ( ) ( ) follows via the transformation (5.3) the differential equation for From (5.10) follows that they obey the following differential equation with a self-adjoint operator in front (5.12) In addition we define now the following set of functions They satisfy the following differential equation ( ) From (3.12) follows their orthonormality and connected with them the completeness relations on the real axis and with the explicit representation Immediately we see that the powers n x of the variable x can be represented (analogously to the case of Hermite polynomials) by application of the reciprocal operator to ( ) 3) The reciprocal functions to ( ) are not explicitly considered up to now in the common literature and thus the practical use of the inversion (6.3) is limited.
For the further investigations it is useful to prepare a few relations for the introduced function From the first relation one obtains the recurrence relation we have interchanging these operators but preserving the commutator only to change the sign of one of these operators, and the operator identity (6.6) can be translated into the operator identity ( ) Operator identities can be applied to arbitrary functions (here correct results in every case.
The result of the differentiation of the polynomials .
Therefore, the operator x ∂ ∂ is the lowering operator for the sets of polynomials The case of the raising operator is more difficult and we delay this problem to Section 9 where we describe the ( ) symmetry of the of the Ultraspherical polynomials ( ) ( ) and of the polynomials . However, applying, for example, the operator identity (6.7) we obtain a partially interesting result which raising the index The operator in front of that means with the definition of ( ) We used the commutation relation x The number of different generating functions which one may calculate in this way is not very large since usually it is not possible to get an explicit result if one applies the operator j x which last one often may not calculate.However, in cases if one has calculated a generating function in another way also for the Ultraspherical polynomials then one may establish a relation to the described way that is usually equivalent to find ordering relations for combinations of functions of operators x and x ∂ ∂ which are difficult to find or to conjecture in another way.

Back Transformation to the Ultraspherical Polynomials and Their Alternative Definition
In this section we make the inverse transformation from the polynomials ( ) If one goes now back from the coordinate x to the coordinate u by relations (5.3) one finds This may be considered as an alternative definition of the Ultraspherical polynomials in comparison to the Rodrigues definition (3.3) (see footnote 2 in Section 5) which on the first glance seems to be awkward but if we look nearer to it we discover also some interesting features and representations, in particular, when we represent this by trigonometric or hyperbolic functions.It is interesting that only the fixed operator ( ) and make the transition from coordinate u to the coordinate x via relations (3) then one finds as the Rodrigues-like definition of ( ) In comparison with the definition (6.1) this definition seems to be more complicated in its structure and the conjecture is that no other substantial different Rodrigues-like definition of the polynomials For the variable u within the borders 1 1 u − ≤ ≤ + we make now the substitution ( ) and obtain from (7.2) For the use of the Ultraspherical polynomials with the variable u in the range 1 u ≥ we can make the substitution by the Hyperbolic Cosine ( ) and in variable ( ) The powers of the operators ( ) One difficulty in last definitions as explained is that the operators ( ) are not "disentangled" as pure series in the differentiation operator θ ∂ ∂ and depend on the degree n of the polynomials.
Here is the right place to mention the following known interesting relations [1] [ ) that leads immediately to (we change temporarily vari- After transformation to the variable u according to (5.3) one finds that after the substitution where in second line we made a Taylor series expansion of ( ) (see (2.5)) one finds the interesting identity ( ) with t as an arbitrary complex parameter.
In the special cases 1 1 , 0, 2 2 α = − this generating function takes on the special forms ( ) At least the second of these generating functions for the Legendre polynomials ( ) n P u is known (see [10], p. 165, Equation ( 5)) but it does not belong to the best-known generating functions and is rarely considered in the corresponding literature.

( )
, su 1 1 In [41] it was proved in a very general way that a quantum-mechanical system with a quadratic spectrum of the energy levels n E as function of n ordered by arising energy possesses a Lie algebra ( ) of lowering and raising operators whereas a linear spectrum as function of n belongs to a Heisenberg-Weyl algebra ( ) (two-dimensional phase space) as it is well known from the harmonic oscillator.If by changing a parameter of the system one has a limiting transition from a quadratic spectrum to a linear spectrum of energy eigenvalues then this transition is expressed by a limiting transition of the corresponding Lie algebras from ( ) . Such a transition is called Inönü-Wigner contraction (see [41]).This procedure is applicable to the Ultraspherical polynomials and their relatives and the orthonormalized functions determined by them.
We may see this from the differential equation for the Ultraspherical polynomials ( ) ( ) Instead of 1 K and 2 K one may introduce the lowering operator K − and the raising operator K + by They obey the commutation relations By the substitutions 1 to the Lie group ( ) but this violates the Hermitecity of the operators ( ) , , L L L if it is supposed for ( ) , , K K K or vice versa.However, within the complex algebra ( )  whereas all unitary irreducible representations of ( ) 1,1 su are (countably) infinite-dimensional with real parameter k which may take on continuous values in a certain range as we will see.One of the first and at once most comprehensive representations of the connection of Special functions to Lie algebras and Lie groups was given by Vilenkin [26] but it is not fully easy to establish the connection to our further representation here and in next Section 5 .
We now discuss in abstract form the unitary infinite-dimensional irreducible representations of ( ) . Only under the very general supposition that there exists a normalized eigenstate , 0 k of the operator 0 K to a real eigenvalue k which is annihilated by the lowering operator − K according to 0 , 0 , 0 , , 0 , 0 1, , 0 0, where k is a fixed parameter of the representation one may construct the whole algebra of ( ) in application to all eigenstates of this (irreducible) representation.One easily sees that the operator K − lowers the eigenvalues by a step of 1 and K + raises the eigenvalues of states by a step of 1.We get in this way new eigenstates , k n of 0 K to eigenvalues n k but widely equivalent to them and defined by These functions are chosen as favorable ones for the representation of the finite-dimensional unitary representations of the Lie group SU(2) following Gel'fand, Naimark et al. [27].The finitedimensional representations of SU(2) become non-unitary (or, more specifically, "quasi-unitary" according to used notion in [26]) for the transition from SU(2) to ( ) ( ) This means that the states , k n can be generated from the lowest state , 0 k according to ) By construction the unitary irreducible representations with parameter k terminate to below with the state to n = 0. Formulae ( 6) agree with corresponding formulae of Van der Jeugt [32] ((1.11) on p. 33).
From (8.6) one may derive the relations .
For the sum and difference of the operator products K K − + and K K + − one finds , , . 2 The operator C defined by ( ) ( ) commutes with all operators of the Lie algebra ( ) and is therefore proportional to the identity operator I of the space of an irreducible representation and is called the Casimir operator for ( ) 1,1 su .The left-hand sides of (8) represents two different factorizations of the main part of the operator of the differential equations.These are, in principle, the Infeld-Hull factorizations [43] and in next Section we find the factors explicitly for the Ultraspherical polynomials.
Abstract equations of the form where we write in braces a sum of Hermitean operator and an anti-Hermitean operator that makes the transition to the raising operator by adjunction easier † u u The first operator ( ) in braces is not independent on n and therefore not a linear operator in usual sense because it depends on the index of the function ( ) ( ) to which it is applied and we have to take this into account and come in rigorous way by inspection of the general relations for the Lie algebra ( ) given in last Section to the raising relation ) and thus this parameter ν can be exactly identified with the parameter k in formulae (8.6).At the end of this Section we will make a remark how one may obtain genuine operators which do not depend on n but this is more a principal action than an action to apply in praxis.
We discuss now the orthonormalized functions  transform it into a form independent of n that, however, we do not make here.
The operation of lowering in (9.4) was written in a form two sum terms where the first is a Hermitean (or self-adjoint) operator and the second an anti-Hermitean operator.By Hermitean conjugation we find from the left-hand side an operator which raise the index from 1 n − to n and if we rise the index n by a step of 1 we find the following operation of raising ( ) The operations (9.1) and (9.4) and also (9.2) and (9.5) or widely analogous and may also be obtained by the coordinate transformation u x ↔ given in (5.3).
In Appendix D we compile these relations using non-Hermitean operators in braces and give the results for the calculations of K K − + and K K + − in the considered realizations of the operations K − and K + .

A. Wünsche
Hermite polynomials although the last are much simpler 6 .
The lowering and raising operators K − and K + in the above relations were obtained in a form where they depend on the number n of the eigenfunctions.Therefore, their application to an arbitrary function is only possible after decomposition of this function into the basis of the orthonormalized eigenfunctions.This can be removed by observing that the operator of the eigenvalue equation is quadratical in the numbers n and is in abstract form of the kind with coefficients ( ) , , a b c and where H is an operator which we call Hamilton operator.We may formally solve the quadratic Equation (9.6) for n with the solution Inserting for n the operator on the right-hand side in the formulae for the lowering and raising operator one obtains it in the general operator form.The sign has to be chosen in such a way that it corresponds to non-negative n .For example, in the realization of ( ) and the Hamilton operator corresponds in this realization to ( ) ( ) . This shows that the lowering and raising operators for ( ) 1,1 su are, principally, well-defined linear operators but for practical purposes their application remains to be difficult.
The action of the annihilation and creation operators onto the orthonormalized Hermite functions ( ) as a realization of the abstract states n can be described by that means with operators on the left-hand side in a form independent of the index n of the functions ( ) n h x .This is connected with the differential equation for the Hermite functions where its Hamilton operator H can be factorized in different way

Generating Functions for Jacobi Polynomials
In this and the following Sections we derive some formulae for general Jacobi polynomials ( ) ( ) . The best-known and basic linear generating function for general Jacobi polynomials is ( [1] and, e.g., [3] . According to citation by Szego [1] (p. 69) it goes back to Jacobi.In the monograph of Srivastava and Manocha [15] besides this linear generating function one may find also a bilinear generating function from Bailey (Equation ( 5) on p.
83) with a function 4 F defined on p. 66 (Equation 26).The problem with the generating function ( 1) is that for α β = it does not make the transition to the most simple basic generation function for the polynomials or Gegenbauer polynomials . Therefore, Rainville [10] (pp.
255, 256) and Andrews et al. [18] ((Equation 6.4.7) on p. 301 in [18]) (see also Koornwinder et al. [17], p. 449, Equation 18.12.3)derive yet another generating function for the Jacobi polynomials for ( In most representations of Orthogonal polynomials this generating function is derived directly from a definition of the polynomials or it is used as basis for the definition of polynomials which generalize the Legendre polynomials and are called then Gegenbauer polynomials (e.g., [10]).This basic generating function alone justifies their independent introduction in comparison to the Ultraspherical polynomials as special case of the Jacobi polynomials with a separate name.
A further generating function for Jacobi polynomials which is remarkable for its symmetry and for its factorization into a product with factors depending only on one of the parameters ( ) , α β is (Koornwinder et al. [17], p. 449, Equation 18.12.2)We direct now our attention to the following generating function for the Jacobi polynomials ( ) ( ) which can be derived from (2.7) as follows (see also [15], p. 82) where we made the substitution n k l − = of the summation indices and made a reordering of the summation.Using the Taylor series for ( ) In contrast to (10.1), the upper indices of the Jacobi polynomials in this generating function are not fixed parameters.
From this one finds the following alternative formula for the polynomials In the special case α β = of ultraspherical polynomials we obtain from (10.6) Apart from bilinear and even trilinear generating functions for Jacobi polynomials we find in Chap. 2 of the monograph of Srivastava and Manocha [15] also more general generating functions from type (10.6)where the upper indices are not pure parameters.Furthermore, we mention that we did not find up to now a generating function for the general Jacobi polynomials which generalizes the generating function (7.16) for the Ultraspherical polynomials.

Operator Ordering and Operator Disentanglement for General Jacobi Polynomials
In some regions of physics, in particular, in quantum optics and in the theory of or the disentanglement of powers of the kind where ( ) ( ) ( ) ( ) ( ) where α and 0 u are arbitrary parameters.We con- jecture from the explicit form of the disentanglement for low numbers n the general relation The correctness for arbitrary n can be proved by complete induction and since this proof is relatively simple, we do not write it down here.
We mention that the operators independent on the value of the parameter β and therefore also for where we made the substitution j k l = + of the summation indices and applied a reordering of the sum terms in the finite sums.By comparison with (2.7) we find that the inner sum can be evaluated according to and the disentanglement relation (11.9) to "normal ordering" can be written in the following final form This is an operator identity which may be applied to arbitrary functions Using the Taylor series of the exponential function we may derive from (11.11) where we changed in an intermediate step the order of summations in the double sum.For the sum on the right-hand side of (11.11) we may apply the generating function (10.6) and we find the operator identity in the form 7 exp 1 1 It is easy to see that this operator identity is correct in its Taylor series in powers of t up to linear terms in t and we checked by computer that this is also the case up to quadratic terms in t .
For an arbitrary function ( ) x ϕ we find in similar way the more general operator identity where ( ) ( ) This or (12) and formula (11) are analogous to corresponding operator identities for Hermite polynomials and thus for the Heisenberg-Weyl group as follows with parameter λ and due to and (11.13) simplifies to ( ) In next section, we apply the derived operator identities to different functions and obtain in this way function identities.

Derivation of Function Identities and a Convolution Identity for Jacobi Polynomials
It was already said that the derived operator identities (11 For example, by applying (11.11) to the functions !n u n one obtains the identity of the following functions on both side Since (12.1) and (12.2) are no more operator identities they cannot be further applied to functions written to the right on both sides.It is easy to check this identity in case of low integers n (we made it up to If we apply the operator identity (11.13) to an arbitrary function ( ) By series expansion of both sides in powers of t one can check this identity for low powers of t (we made it up to powers 2 t ).
The main purpose of this Section is the application of the operator identity (11.11) to the function Relation (12.9) follows then by identification of the coefficients to equal powers n t of the arbitrary t on both sides.In contrast to the derived convolution identities (12.7) the identities (12.9) do not involve the upper indices in the summation and it seems to be not easy to extend (12.9) to more general simple function identities for general Jacobi With the known values of the Ultraspherical polynomials for argument 0 u = given in (4.8) the identity (12.6) takes on the form In special case α β = this leads to the following sum evaluation which is also possible with Vandermond's convolution identity.The special cases 1 u = ± in (12.6) using (2.5) lead also to special cases of Vandermond's convolution identity.One may look to (12.6) as to a generalization of Vandermond's convolution identity for binomial coefficients and the reason for this becomes clear if we take into account that the polynomials ( ) ( )

Conclusion
We derived new functions and representations for Jacobi and for the Ultras- We established a new operator identity for the general case of the Jacobi polynomials which is a kind of operator disentanglement and insofar it is related to reordering of non-commuting operators to normal ordering (all differential operators behind the multiplication operators) and is important and well known in quantum optics for the annihilation and creation operators of the Heisenberg-Weyl group and also in the theory of differential equations.Operator identities can be applied to arbitrary functions and they provide then function identities.
In this way we could prove a kind of convolution theorem for the Jacobi polynomials with a certain similarity to the Vandermond convolution identity for binomial coefficients.Sometimes it was difficult to find out within the immense literature to polynomials whether or not a particular formula or approach is already known or is it novel and our main attention was directed to the correctness of the formulae.
The Jacobi polynomials ( ) ( ) play a main role in the formulae for the Weyl correspondence between classical phase-space functions to quantummechanical operators.This was shortly communicated in [41] and an article to this is in preparation.which is identical with (2.8).In the last step we used the sum evaluation

Appendix B. Evaluation of Two Finite Sums
We prove in this Appendix by the same method the evaluation of two finite sums by expressions of multiplicative type.
First, we prove the correctness of the evaluation of the inner sum in (4.One method for a proof is the following.We transform the sum as follows (the upper limit 2 j of the summation is here extended to ∞ ; intermediately we In the intermediate calculations we used fractional differentiation in case when the parameter α is not an integer.Fractional differentiation and integration is well founded (e.g., [39] [44]).A proof by complete induction from 1 j j → + which, however, is not simple seems also to be possible.
We found that the sum evaluation (B.1) is equivalent to the relations (4.8) and if we now calculate the values for ( ) ( ) The calculated two sums are widely isolated and, apparently, cannot be extended to classes of sums with a continuous parameter, for example, substituting within the sum (1) ( ) → in the sense that these sums may be evaluated also by a formula of multiplicative type.
Table of polynomials ( ) polynomials in comparison to the Rodrigues definition with the treatment of the Ultraspherical polynomials as a realization of the (infinite-dimensional) unitary irreducible representations of the group ( ) 1,1 SU .We determine the lowering and raising operators in this realization of ( ) 1,1 SU by the orthonormalized functions to the Ultraspherical polynomials with fixed upper index as parameter.
weight function of the Jacobi polynomials we find from (2.1) the following useful modification of the definition correspondingly for b instead of a.By a known general transformation relation of the Hypergeometric function where the variable z transforms into the variable 1 z z − ) and (2.18) the parameters ( ) can be chosen within wide borders.
F a b c z without a hint to the possible representation by the Jacobi poly- nomials.

25 ) 2 α and 2 β
The upper indices 1 α > − and 1 β > − are two independent parameters and their squares determine the potential function ( ) U x which are the Pöschl-Teller potentials and the energy eigenvalues depend quadratically on n .The walls at 0 x = and π nomials into a representation by the Gegenbauer polynomials.Furthermore, some of the Ultraspherical polynomials with a low fixed upper index due to their importance in applications got special names and notations.These are the Chebyshev polynomials of first kind (upper index 1

From ( 2 . 1 )
vanishing for 0 n ≠ and are badly appropriate for the definition of the Chebyshev polynomials of first kind ( ) n T u but their definition from the Ultraspherical polynomials is simple and unique.We derive now some of the basic relations for Ultraspherical polynomials ( ) ( ) follows in case of α β = as definition of the kind of the Rodrigues-type )) which transforms the Hypergeometric equation in the sum of two terms containing the Hypergeometric function on the right-hand side from which in special case of the Ultraspherical polynomials one term vanishes for even degree 2 n m = and the other term for odd degree 2 1 n m =+ and one hase the two separate relations[1] are also Ultraspherical polynomials.We mention yet in this connection that due to the transformation relation (2.30) which changes one of the upper indices and which for α β = takes on the form powers of products of x and y and the 3 Although the obtained explicit representation of the Ultraspherical polynomials (4.4) is known from other more indirect approaches it is interesting to make this sum evaluation directly in (4.3), moreover, since in next Section we encounter another sum which can be evaluated in similar way.representation of the coefficients by the Jacobi polynomials as special case

7 )
or if we use the standard notation for the Bessel functions ( )

10 )
They are not orthogonal polynomials as we will see.Instead of the polynomials

. 6 .
After these preparations it is necessary here to mention the following.The sets of polynomials α are not sets of Orthogonal poly- nomials in the usual sense because for their definition we cannot find a weight function depend on the degree n of the polynomials but may depend only on α as parameter but instead of this the sets of func- to the Classical orthogonal polynomials but are relatives to the Ultraspherical polynomials ( ) ( ) Investigation of the Relatives to the UltrasphericalPolynomials and a Generating FunctionWe continue in this section the investigation of the relatives ( )

9 )
This is not a pure raising relation for the polynomials second part on the right-hand side we have yet a lowering of the index n accompanied with the raising of the upper index α .Another approach to the problem of the raising operator leads only to a form which is formally dependent on the degree n of the polynomial and we make this in Sections 8 and 9 for the normalized polynomials

α
on the left-hand side may be seen as a form of the creation operator, which however depends on the index n of n G α to which it is applied and must be changed if it is applied to a polynomial with changed degree n of the polynomials plays a main role in application to the to make the opposite to go from the Ultraspherical polynomials

9 )
They lead to the Chebyshev polynomials of first and second kind 10) in comparison to the Rodrigues definition ( ) 13) with such simple relations to the algebra of trigonometric functions but it does not belong to the Ultraspherical polynomials since their upper indices are not equal.Let us now translate the generating function (13) for ( ) with the dependence on n is a quadratic one.With the parts which are independent on n in these differential equations one may determine an analogue to the quantum-mechanical Hamilton operator in the Schrödinger equation.The same is true for the discussed relatives n is a quadratical one.The explicit form of the operators of lowering and raising the indices in their concrete realization depends on the definition of the functions as normalized functions or as otherwise ones and only for the orthonormalized functions they can be defined in an abstract general form because all other forms are hardly to characterize by some common properties.This abstract form for the discrete representations of the Lie algebra [42] and was given in many of our later papers.To prepare the special treatment for the Ultraspherical polynomials and for their relatives we compile first shortly and without proof the relations in the abstract form.Lie algebra and possesses as basis three (by definition) Hermitean or self-adjoint basis operators ( ) abstract way by the commutation relations

9 ., SU 1 1 α
to the eigenvalue equations in concrete realizations of the states , k n by functions.Using then (8) which are also equivalent to the eigenvalue equations one obtains different factorizations K K − + and K K + − of the main operator part (Hamilton operator) in these equations.Lowering and Raising Operations for the Ultraspherical Polynomials and Their Relatives in Connection with ( ) -Symmetry In the treatment of the problem of algebra of lowering and raising operators for the Ultraspherical polynomials ( ) ( ) such, for example, for the polynomials and then one can make the transition to the orthonormalized functions.If one has, for example, the lowering operator then one can make the transition to the raising operator by transition to the adjoint operator.In both cases of the orthonormalized functions is given in (3.14) and the raising operator is also known.Starting from the relation for lowering in (3.14) one may make in straight-forward way the transition to the normalized functions ( ) ( ) in (3.10) and find a lowering relation which can be written in the form

2 )
By comparison of the right-hand sides of (9.1) and (9.2) with the right-hand sides in (8.6) one finds the connection of the parameter α in ( ) ( ) k in the abstract relations to be 13) the lowering relation which we may write using self-adoint operators in braces in the following form lowering relation(9.4)    must possess the principal form given in(8.6)and this is only possible if we identify the parameter k with 1 2 α + such as given in (9.3).The left-hand side of the operations has not yet a form independent on the number n of the function differential equations it should be possible to annihilation operator a and creation operator † a obey the commutation relation of a Heisenberg-Weyl algebra ( ) 2 w and applied to abstract orthonormalized states n act by † † † ,

2 )
which, however, in case of α β = makes the transition to the following relatively simple basic generating function for Ultraspherical polynomials (Equation (6.4.8) in [18]) This generating function becomes particularly simple in terms of the Gegenrelation (3.1) to the Ultraspherical polynomials ( ) ( ) differential equations it is important to have for disposition some ordering relations of non-commuting operators.To illustrate this we give some simple and known examples.For the operators of differentiation x for a Heisenberg-Weyl Lie algebra the normal ordering is particularly important.Normal ordering is if all functions of the multiplication operator x are in front of powers of the differentiation operator x ∂ ∂ Such relations are, for example, the transition from anti-normal ordering to normal ordering for powers of both operators

4 )
S n k denotes the Stirling numbers of second kind which together with the Stirling numbers of first kind ( ) , s n k obey the orthonormality relation (e.g., [45] [46]) This relation makes it possible to find the conversion of the last operator identities.As a rule such operator identities can be proved by complete induction in connection with recurrence relations.We now derive operator identities which arise specifically in connection with the theory of Jacobi polynomials.First we observe in connection with the weight function ( bring it into a form where all powers of the differentiation operators u ∂ ∂ are to the right of functions ( ) f u of the multiplication operator u .We begin with the solution of the partial problem for the operator the operator disentanglement formulae(11.11)takes on the form 7

=
for operators A and B if

[
representation of linear transformations in products of binomials.
pherical polynomials and of their special cases of Legendre polynomials and of Chebyshev polynomials of first and of second kind.A main aim was to find for the Jacobi polynomials an alternative definition in comparison to the Rodrigues definition but in analogy to the alternative definition of Hermite polynomials (see footnote in Section 5) where this alternative definition was very successful and led in the past also to a basic definition of the Laguerre 2D and Hermite 2D polynomials.This last has shown the way for a unification of Hermite and Laguerre polynomials on a common two-dimensional level with two-dimensional transformation relations between them and, interestingly, had essentially to use the Jacobi polynomials for their representation but not with fixed upper indices as parameters as mainly in present work.We could realize this programme only for the Ultraspherical polynomials as special case of the Jacobi polynomials.A key to a possible generalization of the approach presented in this article is to find a relation which generalizes in relation (6.1) the polynomials ( ) n G x α to more general polynomials which depend on parameters α and β A. Wünsche and which are generated from the monomials nx by an operator depending only the parameters ( ) , α β but not on the degree n of the polynomials.From such a relation as (6.1) if it exists it is then necessary to make the transformation back to the Jacobi polynomials.Such alternative definitions possess the advantage that we may generate the polynomials of n from the monomials n x by a common operator and many calculations can be separated then into corresponding calculations for the monomials n x and after this to the appli- cation of the common operator.The auxiliary set of polynomials differentiation relation for them derived in this article are also interesting for their own.A further aim was to establish explicitly the group-theoretical background of Ultraspherical polynomials in form of the the differential equations which besides the operator part including derivatives up to the second order contains an eigenvalue part which is quadratically dependent on the degree n of the polynomials.In an earlier paper we proved in very general way that a Hamilton operator with quadratical energy spectrum n E in powers of n has ( ) 1,1 SU as background and in case of a linear spectrum this is the Heisenberg-Weyl group ( ) 2 W .The factual establishment of the relations of the algebra of lowering and raising operators of the Jacobi polynomials to ( ) 1,1 SU can be made via the corresponding orthonormalized functions.Surely, this part can be generalized to the whole sets of Jacobi polynomials (e.g., general Pöschl-Teller potentials) but was explicitly made in present article only to its special case of Ultraspherical polynomials.
the expansion (2.7) one finds that this is equivalent to the sum evaluations checked by computer but not independently proved.Second, we prove the correctness of the evaluation of the inner sum in(5

+
and the relations to Chebyshev polynomials of first and second kind ( ) ( ) , n n T u U u and Legendre polynomials