Abstract

In this paper, we derive eight basic identities of symmetry in three variables related to q-Euler polynomials and the q -analogue of alternating power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundance of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on the p-adic integral expression of the generating function for the q -Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the q -analogue of alternating power sums.

1. Introduction and Preliminaries

Let p be a fixed odd prime. Throughout this paper, , , will respectively denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure of. For a continuous function, the p-adic fermionic integral of f is defined by

Then it is easy to see that

(1)

Let be the normalized absolute value of, such that, and let

(2)

Assume that, with, so that and are, as functions of z, analytic functions on. By applying (1) to f, with, we get the p-adic integral expression of the generating function for q-Euler numbers:

(3)

So we have the following p-adic integral expression of the generating function for the q-Euler polynomials:

(4)

Note here that in [7] was used in place of q, and that q -Euler numbers and polynomials were coined respectively as -Euler numbers and polynomials.

Let denote the q-analogue of alternating kth power sum of the first n + 1 nonnegative integers, namely

(5)

In particular,

(6)

From (3) and (5), one easily derives the following identities: for any odd positive integer w,

(7)

In what follows, we will always assume that the p-adic fermionic integrals of the various exponential functions on are defined for (cf. (2)), and therefore it will not be mentioned.

[1,2,5,8,9] are some of the previous works on identities of symmetry involving Bernoulli polynomials and power sums. These results were generalized in [4] to obtain identities of symmetry involving three variables in contrast to the previous works involving just two variables.

In this paper, we will produce 8 basic identities of symmetry in three variables w₁, w₂, w₃ related to q-Euler polynomials and the q-analogue of alternating power sums (cf. (44), (45), (48), (51), (55), (57), (59), (60)). These and most of their corollaries seem to be new, since there have been results only about identities of symmetry in two variables in the literature. These abundance of symmetries shed new light even on the existing identities. For instance, it has been known that (8) and (9) are equal and (10) and (11) are so (cf. [7,(2.11),(2,16)]). In fact, (8)-(11) are all equal, as they can be derived from one and the same p-adic integral. Perhaps, this was neglected to mention in [7]. In addition, we have a bunch of new identities in (12)-(15). All of these were obtained as corollaries(cf. Cor. 4.9, 4.12, 4.15) to some of the basic identities by specializing the variable w₃ as 1. Those would not be unearthed if more symmetries had not been available. Related to q-Bernoulli polynomials and the q-analogue of power sums, identities of symmetry in three variables were also obtained in [3] as an extension of identities of symmetry in two variables in [6].

Let w₁, w₂, be any odd positive integers. Then we have:

(8)

(9)

(10)

(11)

       

(12)

(13)

(14)

(15)

The derivations of identities are based on the p-adic integral expression of the generating function for the q-Euler polynomials in (4) and the quotient of integrals in (7) that can be expressed as the exponential generating function for the q -analogue of alternating power sums. We indebted this idea to the papers [5,6].

2. Several Types of Quotients of Fermionic Integrals

Here we will introduce several types of quotients of p-adic fermionic integrals on or from which some interesting identities follow owing to the built-in symmetries in w₁, w₂, w₃. In the following, w₁, w₂, w₃ are all positive integers and all of the explicit expressions of integrals in (17), (19), (21), and (23) are obtained from the identity in (3).

(a) Type (for)

(16)

(17)

(b) Type (for)

(18)

(19)

(c-0) Type

(20)

(21)

(c-1) Type

(22)

(23)

All of the above p-adic integrals of various types are invariant under all permutations of w₁, w₂, w₃, as one can see either from p -adic integral representations in (16), (18), (20), and (22) or from their explicit evaluations in (17), (19), (21), and (23).

3. Identities for q-Euler Polynomials

In the following w₁, w₂, w₃, are all odd positive integers except for (a – 0) and (c – 0), where they are any positive integers.

(a – 0) First, let’s consider Type, for each The following results can be easily obtained from (4) and (7).

(24)

                 

(24)

where the inner sum is over all nonnegative integers k, l, m, with, and

(25)

(a-1) Here we write in two different ways:

1)      (26)

         

(27)

2) Invoking (7), (26) can also be written as

(28)

  

(a-2) Here we write in three different ways:

1)

(29)

         

(30)

2) Invoking (7), (29) can also be written as

(31)

(32)

3) Invoking (7) once again, (31) can be written as

 

(33)

(a-3)

             

            

(34)

b) For Type, we may consider the analogous things to the ones in (a-0), (a-1), (a-2), and (a-3). However, these do not lead us to new identities. Indeed, if we substitute respectively for in (16), this amounts to replacing t by and q by in (18). So, upon replacing respectively by, and then dividing by and replacing by q, in each of the expressions of Theorem 4.1 through Corollary 4.15, we will get the corresponding symmetric identities for Type.

(c-0)

               

(35)

(c-1)

              

              

(36)

4. Main Theorems

As we noted earlier in the last paragraph of Section 2, the various types of quotients of p-adic fermionic integrals are invariant under any permutation of. So the corresponding expressions in Section 3 are also invariant under any permutation of. Thus our results about identities of symmetry will be immediate consequences of this observation.

However, not all permutations of an expression in Section 3 yield distinct ones. In fact, as these expressions are obtained by permuting in a single one labeled by them, they can be viewed as a group in a natural manner and hence it is isomorphic to a quotient of. In particular, the number of possible distinct expressions are 1, 2, 3, or 6. (a-0), (a-1(1)), (a-1(2)), and (a-2(2)) give the full six identities of symmetry, (a-2(1)) and (a-2(3)) yield three identities of symmetry, and (c-0) and (c-1) give two identities of symmetry, while the expression in (a-3) yields no identities of symmetry.

Here we will just consider the cases of Theorems 4.8 and 4.17, leaving the others as easy exercises for the reader. As for the case of Theorem 4.8, in addition to (50)-(52), we get the following three ones:

(37)

(38)

(39)

But, by interchanging l and m, we see that (37), (38), and (39) are respectively equal to (50), (51), and (52).

As to Theorem 4.17, in addition to (60) and (61), we have:

(40)

(41)

(42)

(43)

However, (40) and (41) are equal to (60), as we can see by applying the permutations for (40) and for (41). Similarly, we see that (42) and (43) are equal to (61), by applying permutations for (42) and for (43).

Theorem 4.1 Let be any positive integers. Then we have :

(44)

Theorem 4.2 Let be any odd positive integers. Then we have:

(45)

     

     

                

     

Putting in (45), we get the following corollary.

Corollary 4.3  Let be any odd positive integers. Then we have:

(46)

Letting further in (46), we have the following corollary.

Corollary 4.4 Let be any odd positive integer. Then we have:

             

(47)

Theorem 4.5 Let be any odd positive integers. Then we have:

(48)

        

Letting in (48), we obtain alternative expressions for the identities in (46).

Corollary 4.6 Let be any odd positive integers. Then we have:

(49)

           

            

Putting further in (49), we have the alternative expressions for the identities for (47).

Corollary 4.7 Let be any odd positive integer. Then we have:

Theorem 4.8 Let be any odd positive integers. Then we have:

(50)

(51)

(52)

Putting in (50)-(52), we get the following corollary.

Corollary 4.9 Let be any odd positive integers. Then we have:

     

(53)

Letting further in (53), we get the following corollary. This was also obtained in [7,(2.12)].

Corollary 4.10 Let be any odd positive integer. Then we have:

(54)

Theorem 4.11 Let be any odd positive integers. Then we have:

          

(55)

Putting in (55), we obtain the following corollary. In Section 1, the identities in (53), (56), and (58) are combined to give those in (8)-(15).

Corollary 4.12 Let be any odd positive integers. Then we have:

(56)

Letting further in (56), we get the following corollary. This is the identity obtained in (54) together with the multiplication formula for q-Euler polynomials [cf.7,(2.17)].

Corollary 4.13 Let be any odd positive integer. Then we have:

     

Theorem 4.14 Let be any odd positive integers. Then we have:

               

              

(57)

Letting in (57), we have the following corollary.

Corollary 4.15 Let be any odd positive integers. Then we have:

(58)

Theorem 4.16 Let be any positive integers. Then we have:

(59)

Theorem 4.17 Let be any odd positive integers. Then we have:

(60)

(61)

Putting in (60) and (61), we get the following corollary.

Corollary 4.18 Let be any odd positive integers. Then we have:

5. References

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[2] F. T. Howard, Applications of a recurrence for the Bernoulli numbers, J. Number Theory 52(1995), 157-172. doi:10.1006/jnth.1995.1062

[3] D. S. Kim, Identities of symmetry for q-Bernoulli polynomials, submitted. doi:10.1080/10236190801943220

[4] D. S. Kim and K. H. Park, Identities of symmetry for Bernoulli polynomials arising from quotients of Volkenborn integrals invariant under S₃, submitted.

[5] T. Kim, Symmetry p-adic invariant integral on for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14 (2008), 1267-1277.

[6] T. Kim, On the symmetries of the q-Bernoulli polynomials, Abstr. Appl. Anal. 2008(2008), 7 pages(Article ID 914367).

[7] T. Kim, K. H. Park, and K.W. Hwang, On the identities of symmetry for the ζ-Euler polynomials of higher-order , Adv. Difference Equ. 2009(2009), 9 pages (Article ID 273545).

[8] H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), 258-261. doi:10.2307/2695389

[9] S. Yang, An identity of symmetry for the Bernoulli polynomials, Discrete Math. 308 (2008), 550-554. doi:10.1016/j.disc.2007.03.030