Generalized Exponential Euler Polynomials and Exponential Splines ()
Here presented is constructive generalization of exponential Euler polynomial and exponential splines based on the interrelationship between the set of concepts of Eulerian polynomials, Eulerian numbers, and Eulerian fractions and the set of concepts related to spline functions. The applications of generalized exponential Euler polynomials in series transformations and expansions are also given.
1. Introduction
Eulerian polynomial is defined by the following equation (cf. for examples, [1-3]):
(1.1)
It is well-known that the Eulerian polynomial, 
, of degree 
can be written in the form
(1.2)
where 
are called the Eulerian numbers that can be calculated by using
(1.3)
The Eulerian number 
gives the number of permutations of 
with 
permutation ascents (cf. [4]), or equivalently, the number of permutation runs of length 
(cf. [3]), where a set of ascending sequences in a permutation is called a run and 
is a permutation ascent if the 
th term in the permutation less than the 
st term. A geometric interpretation of Eulerian number 
were presented in [5] and [6], where which was found equaling to the volume of 
th slice of a 
-dimensional unit cubenamely,
. It is wellknown that the Eulerian fraction is a powerful tool in the study of the Eulerian numbers, Eulerian polynomial, Euler function and its generalization, Jordan function ( cf. [7]). The classical Eulerian fraction, 
, can be expressed in the form
(1.4)
spline functions are probably the most applicable one-dimensional polynomial spline functions. The values of 
spline functions can be defined as the volumes of slices of a 
-dimensional cube (cf. [8]). Hence, one can expect a tight connection between the Eulerian numbers and the 
spline function values. This is is one of the motivations to study the interrelationship between Eulerian polynomials and B-splines. Another motivation is from the construction of the exponential splines by using the exponential Euler polynomial discussed in [9], which will be described in this paper.
Polynomial spline functions can be considered as broken polynomials with certain smoothness, which are used to overcome the stiffness of polynomials, for instance the Runge phenomenon. One way to construct the univariate polynomial B-splines is using divided differences to truncated powers, and the truncated powers can be generated by iterated integrations. An extension called the truncated Tchebycheff functions can be constructed similarly using the iterated integrations with respect to positive, smooth weight functions, and the Tchebycheffian B-splines are linear combination of these truncated Tchebycheff functions. Exponential splines belong to a class of Tchebycheffian B-splines, which preserve many features of the polynomial B-splines such as smoothness, compact support, non-negativity, and local linear independence. One may find more details in [7,10,11] and [12] and some applications in [13] and [14]. In [15], Dyn and Ron considered periodic exponential B-splines defined by weight functions 
with
, 
, and showed these B-splines possess a significant property of translation invariant and satisfy a generalized Hermite-Genocchi formula. Motivated by this work, Ron define the higher dimensional 
-directional exponential box splines in [16].
Another approach to construct the exponential splines to the base 
is using a linear combination of the translations of a polynomial B-spline with combination coefficients
. The exponential splines to the base 
can be presented and evaluated by using exponential Euler polynomials, which connect to some famous generating functions (GF’s) in the combinatorics. The details can be found in Lecture 3 of [9-11], and Chapter 2 of [17], and we shall briefly quote them in next section. This approach can be used to derive a strong interrelationship between two different fields. For instance, in [18], the author presents the equivalence between exponential Euler polynomials, Euler-Frobenius polynomials, etc. and some frequently used concepts in combinatorics such as Eulerian fractions, Eulerian polynomials, etc., respectively, which will be surveyed as follows.
Let 
be an integer,
. The symbol 
denotes the class of splines of order
, i.e., functions 
satisfying the following conditions: 1) 
and 2) 
in each interval
,
where 
is the class of polynomials of degree not exceeding
.
In the central part of Schoenberg’s lectures on “Cardinal Spline Interpolation” (CSI) (cf. [9]), the exponential Euler polynomial of degree 
to the base 
is defined as
(1.5)
where
is the exponential spline of degree 
to the base
, where
denote the cardinal forward B-splines. Here, an exponential spline 
means an element in 
satisfying the functional equation
(1.6)
Obviously,
. Therefore, 
, and it is easy to find 
satisfies (1.6).
From [9], 
satisfies 
and is a polynomial in the interval 
with the form
for
. Hence, 
is a monic polynomial in
. From [9], the generating function of 
is
(1.7)
Using (1.7) we may expand both sides of
and comparing the coefficients of power of 
in the expansions on two sides, we obtain
(1.8)
If
, then (1.7) reduces to
(1.9)
where
. It is easy to see that
(1.10)
In particular,
(1.11)
Hence, we call 
the coefficient (polynomials) of the exponential Euler polynomial
.
In addition, multiplying (1.9) by 
and comparing the coefficients of the powers of
, we have the relations
(1.12)
Jointing (1.11) and (1.12) yields
(1.13)
If 
for all integer
, then (1.8) implies
.
Thus, we have
(1.14)
for all integer
.
In the following, we call
(1.15)
the Euler-Frobenius polynomial. Since
(1.16)
defined by (1.15) can be written as
(1.17)
From (1.17), we know 
has 
zeros, all negative and simple, with 
a zero if and only if 
is. A proof can be found, for example, in [17].
On the other hand, from (1.9) and (1.15) we have
Transferring 
to
, we change the above equation as
The left-hand side of the above equation can be expanded as
Comparing the right-hand sides of the last two equations yields
(1.18)
(1.18) implies
(1.19)
In the introduction, we give the following definition of the Eulerian polynomial of degree
, which is similar to (1.18).
(1.20)
where the Eulerian polynomial, 
can be written in the form of (1.2) with the Eulerian numbers calculated using (1.3).
In addition, Eulerian fraction, denoted by
, is defined by
(1.21)
Comparing (1.18) and (1.20), (1.17) and (1.7), and (1.15) and (1.3), [18] gives the following results.
Proposition 1.1 [18] Let 
and 
be the polynomials defined by (1.15) and (1.9), respectively, and let Eulerian polynomials
, Eulerian numbers
, and Eulerian fractions 
be defined by (1.20), (1.2), and (1.21), respectively. Denote the cardinal forward B-spline of order 
by
. Then we can set the relationship between the concepts as
(1.22)
(1.23)
(1.24)
As examples of the interrelationships shown in (1.22) - (1.24), we see
1) 
2) 
3) 
etc.
and
1) 
2) 
3)
; etc.
Using Proposition 1.1, [18] derives many properties of B-spline and Euler-Frobenius polynomials readily from the corresponding properties of Eulerian polynomials and Eulerian numbers, and vice versa. Applications in B-spline interpolation and evaluation of Riemann Zeta function values at odd integers are also given in [18].
Let
. It is well-known that the Eulerian fraction defined by (1.21) can be written as
(1.25)
where 
are the Stirling numbers of the second kind, i.e., 
or equivalently, the number of partitions of 
distinct elements in 
blocks.
The connection shown in Proposition 1.1 also provides some important applications, shown in Section 2, of B-splines and exponential B-splines to the enumerative combinatorics. In the section, a generalized Eulerian fraction and a generalized exponential spline to the base 
are derived. Section 3 discusses the applications of the generalized exponential Euler polynomials in the series transformations and expansions. Furthermore, using the combinatorial technique, we may generate exponential splines in higher dimensional setting, which will be briefly described at the end of the section and will be discussed in a later paper on the multivariate spline functions.
2. Generalized Exponential Splines
In this section, we will introduce a generalized Eulerian fraction, from which a generalized exponential spline to the base 
will be derived. We shall also discuss the corresponding generalized Euler-Frobenius polynomials and their applications.
It is obvious that (1.25) is equivalently
(2.1)
where
. Hence, we may extend the Eulerian fraction to a more general case, which will be denoted by 
and called generalized Eulerian fractions associated with an infinitely differentiable function
:
(2.2)
Therefore,
. An analogue of the coefficients of the exponential Euler polynomials 
defined by (1.10) is extended to
(2.3)
which is associated with an analytic function
.
We now find the generating function of
and
.
Theorem 2.1 Let 
be an infinitely differentiable function, and let 
and 
be defined by (2.2) and (2.3), respectively. Then their exponential generating functions are
(2.4)
and
(2.5)
where
.
Proof. Substituting (2.2) into the right-hand side of (2.4) and using the generalized Euler series transform formula (cf. [19] and (3.1) in [20]), we can evaluate the exponential generating function of 
formally as
Similarly, by substituting (2.3) into the right-hand side of (2.5) and applying formula (2.4) to the resulting expression, we have
which yields (2.5) and completes the proof of the theorem.
From 
we now construct polynomials
for 
using the following expansion.
(2.6)
Thus, 
can be evaluated from 
as follows.
(2.7)
for
. In particular, 
and
We call 
the generalized exponential Euler polynomial of degree 
associated with 
to the base
, and 
the coefficient of
. In addition, we call 
the generalized Euler polynomials, and
, the regular Euler polynomials, are the special case of 
for
.
A generalized exponential spline function to the base
, denoted by
, is therefore defined using 
(
) as follows.
(2.8)
in
. However, if one wants to extend 
to all real 
associated with 
such that
, 
needs to satisfy some periodic conditions. Let us consider the class of
with real constants 
and
.
Then we have the following result.
Theorem 2.2 Let function 
be defined by (2.8) on 
associated with
where 
and 
are real constants and
, 
can be extended to a 
continuous function for all real 
if and only if
. Furthermore, the extension is fulfilled by the functional equation
(2.9)
Proof. From [9], for (2.9) combined with the condition that 
require
to satisfy in 
the periodic conditions
(2.10)
for
. From (2.7), we have
Hence, condition (2.10) is equivalent to
(2.11)
for all
. From (2.7) (2.3), and (2.2), we have
(2.12)
where the last step is due to the identities
and
The double sum in the rightmost expression of (2.12) can be further changed to
where the last step comes from 
for all
. Substituting the last expression of the double sum
into equation
which comes from (2.12), and solving the resulting equation for
, we obtain
(2.13)
Substituting the last equation to the first equation of (2.12) yields
where the last step is implied by
(see (2.2) and (2.3). Therefore, if
, then
. The sufficiency is obvious. This completes the proof of the theorem.
Remark 3.1 Other generalized exponential splines are planed to be constructed. And the properties of the exponential splines need to be investigated.
3. Applications in Series Transformations and Expansions
All formulas presented in this section are formal identities in which we always assume that 
or 
according as 
or 
appears in the formulas.
Theorem 3.1 Let 
and 
be infinitely differentiable on
, and let 
(
) be the generalized exponential Euler polynomial to the base 
defined by (2.6) or (2.7). Then we have formally
(3.1)
where 
is not any poles of 
and 
for any
.
Proof. First, we apply the operator 
to the infinitely differentiable function 
at
, where 
is the shift operator.
On the other hand, we can present
By applying (3.1) in [19] to the inner sum of the rightmost side of the above equation for 
and noting
, we obtain
which implies (3.1) by writing
formally, where for 
and 
we use (2.2) and (2.3), respectively. This completes the proof of the theorem.
Remark 4.1 The series transformation formulas (3.1) have numerous applications by setting different infinitely differentiable functions for
. For instance, If
, then (3.1) becomes to
(3.2)
where 
and the relation (1.5) has been used.
Evidently, when 
is a polynomial, formulas (3.1) and (3.2) become closed form of summation formulas with a finite number of terms. Moreover, the Right-hand side of each formula may also be viewed as a 
for the sequence of coefficeients contained in the power series on the left-hand side. For example, if
, noting
we have
At the end of this section, we will show a way to extend our construction of exponential splines to the higher dimensional setting. First, we shall adopt the multi-index notational system. Denote
Moreover, we write 
with 
being non-negative integers. Also, 
means 
, and 
means 
for all
.
We now define a generalized higher dimensional exponential polynomials 
as follows.
Definition 3.2 Let 
be any given formal power series over the complex number field 
with
.
Then 
defined by
(3.3)
are called the generalized higher dimensional exponential Euler polynomials of degree 
associated with 
to the base
, where
. In particular, for
, (3.3) defines higher dimensional exponential Euler polynomials 
of degree 
to the base 
by
(3.4)
Let
. Then the polynomial 
defined by (3.4) is called the higher dimensional (regular) Euler polynomial of degree
. The more details will be given in a later paper.
4. Acknowledgments
The author greatly appreciate the discussion with Carl deBoor since 2005 and the valuable suggestions and the lecture [17] he offered fervently.
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