Degenerate binomial coefficients and degenerate hypergeometric functions

In this paper, we investigate degenerate versions of the generalized pth order Franel numbers which are certain finite sums involving powers of binomial coefficients. In more detail, we introduce degenerate generalized hypergeometric functions and study degenerate hypergeometric numbers of order p. These numbers involve powers of λ-binomial coefficients and λ-falling sequence, and can be represented by means of the degenerate generalized hypergeometric functions. We derive some explicit expressions and combinatorial identities for those numbers. We also consider several related special numbers like λ-hypergeometric numbers of order p and Apostol type λ-hypergeometric numbers of order p, of which the latter reduce in a limiting case to the generalized pth order Franel numbers.


Introduction
First, we study certain finite sums involving powers of binomial coefficients which are called generalized pth order Franel numbers and can be represented in terms of hypergeometric functions. Then, among other things, we find that particular cases of these numbers are connected with many known special numbers and polynomials which include Bernoulli numbers, Euler numbers, Changhee numbers, Daehee numbers, Stirling numbers of the first kind, Catalan numbers, and Legendre polynomials.
In recent years, many mathematicians have devoted their attention to studying various degenerate versions of some special numbers and polynomials [4,7,8,10,15,16]. The idea of investigating degenerate versions of some special numbers and polynomials originated from Carlitz's papers [2,3]. Indeed, he introduced the degenerate Bernoulli and Euler polynomials and numbers, and investigated some arithmetic and combinatorial aspects of them. Here we mention in passing that the degenerate Bernoulli polynomials were later rediscovered by Ustinov under the name of Korobov polynomials of the second kind. Two of the present authors, their colleagues, and some other people have studied quite a few degenerate versions of special numbers and polynomials with their interest not only in combinatorial and arithmetic properties but also in differential equations and certain symmetric identities (see [9,16] and the references therein). It is worth noting that this idea of considering degenerate versions of some special polynomials and numbers is not only limited to polynomials but can also be extended to transcendental functions like gamma functions [11,12]. We believe that studying some degenerate versions of special polynomials and numbers is a very fruitful and promising area of research in which many things remain yet to be uncovered.
Recently, Dolgy and Kim gave some explicit formulas of degenerate Stirling numbers associated with the degenerate special numbers and polynomials. Motivated by Dolgy and Kim's paper [4], we would like to investigate degenerate versions of the generalized pth order Franel numbers. In more detail, we introduce degenerate generalized hypergeometric functions and study degenerate hypergeometric numbers of order p. These numbers involve powers of λ-binomial coefficients and λ-falling sequence, and can be represented by means of the degenerate generalized hypergeometric functions. We also consider several related special numbers like λ-hypergeometric numbers of order p and Apostol type λ-hypergeometric numbers of order p, of which the latter reduce in a limiting case to the generalized pth order Franel numbers.
For the rest of this section, we will fix some notations and recall some known results that are needed throughout this paper.
The Stirling numbers of the first kind are defined as [13,15] .
Thus, by (7), we get In view of (4), the degenerate Stirling numbers of the first kind are defined by Note that lim λ→0 S 1,λ (n, k) = S 1 (n, k) (n, k ≥ 0). As is well known, the generalized hypergeometric function F (p,q) is defined by where a k = a(a + 1) · · · (a + (k -1)) (k ≥ 1), a 0 = 1 (see [17,21]). For example, The Gauss summation theorem is given by where From (11), we note that where The following are well-known identities related to the binomial coefficients: . (17)

Sums of powers of λ-binomial coefficients
The λ-binomial coefficients are defined as From (18), we easily get By (1), we easily get where n and k are positive integers.
From (21), we note that where n is a nonnegative integer. Let us define Therefore, by (22) and (23), we obtain the following theorem.
We note that lim λ→0 Q λ (m, 2) = n k=0 n k 2 k m = Q(m, 2), which was introduced by Golombek and Marburg (see [5,6]). We observe that For n ∈ N, let On the one hand, we have On the other hand, we get From (24), (25), and (26), we obtain the following theorem.
As is well known, the degenerate Bell polynomials are defined by Bel n,λ (x) t n n! see [16] .
By (27), we easily get Now, we define the degenerate bivariate Bell polynomials by Bel n,λ (x, y) t n n! .
By (33), we get Let us take m = 3. Then we have Note that For s ∈ C with R(s) > 0, the gamma function is defined by Let n be a nonnegative integer. Then From (35), we note that In particular, for z = 1 λ (λ = 0), from (11) we get For n ∈ N, by (37), we get F λ -n, -n λ where λ is a positive real number. On the other hand, Therefore, by (38) and (39), we obtain the following theorem.
For λ, λ 1 ∈ R, let us define Apostol type alternating λ-hypergeometric numbers of order p by investigated degenerate versions of the generalized pth order Franel numbers. In more detail, we introduced degenerate generalized hypergeometric functions and studied degenerate hypergeometric numbers of order p. We showed that the degenerate hypergeometric numbers of order p involve powers of λ-binomial coefficients and λ-falling sequence, and can be represented by means of the degenerate generalized hypergeometric functions. We also considered several related special numbers like λ-hypergeometric numbers of order p and Apostol type λ-hypergeometric numbers of order p, of which the latter reduce in a limiting case to the generalized pth order Franel numbers.