Two types of hypergeometric degenerate Cauchy numbers

Abstract In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first one, we give their expressions and determine the coefficients. Concerning the second one, called H-degenerate Cauchy polynomials, we show several identities and study zeta functions interpolating these polynomials.


Introduction
In 1985, Howard [1, (3.6)] introduced the polynomial ( ) ( ) β λ x , The degenerate Bernoulli polynomials in λ and x have rational coefficients. When x = 0, β n (λ) = β(λ, 0) are called degenerate Bernoulli numbers. In [3], explicit formulas for the coefficients of the polynomial β n (λ) are found. In [4], a general symmetric identity involving the degenerate Bernoulli polynomials and the sums of generalized falling factorials are proved. In [5], a kind of generalization of degenerate Bernoulli numbers, called hypergeometric degenerate Bernoulli numbers, is introduced and its properties are studied. The classical Bernoulli numbers may be called Bernoulli numbers of the first kind too. It is because there exists another kind of Bernoulli numbers, that is, the Bernoulli numbers of the second kind b n , defined by Bernoulli numbers of the second kind b n are often studied as Cauchy numbers (of the first kind) c n . As seen in [3, Lemma 2.1], the following relation holds.
n n (6) It is natural that Cauchy numbers are recognized by t being replaced by log(1 + t) and taking its reciprocal in the generating function in (4)  On the contrary, Bernoulli numbers can be recognized by t being replaced by e t − 1 and taking its reciprocal in the generating function of Cauchy numbers. In fact, log(1 + t) and e t − 1 are inverse functions of each other. Therefore, it is natural to introduce the degenerate Cauchy numbers γ n (λ) by replacing t by log(1 + t) and taking its reciprocal in (2). In fact, in the same paper, as the counterpart of degenerate Bernoulli polynomial, Howard [1, (3.7)] introduces another polynomial ( ) ( ) γ λ x , (see also [6, (2.13)]). This matches our idea about the relation between Bernoulli and Cauchy polynomials by replacing t by log(1 + t) and taking its reciprocal in (2). It seems that this type of degenerate Cauchy polynomials has not been well studied but forgotten. Unfortunately, there seem to exist another definition of degenerate Cauchy polynomials without any historical background or motivation.

Degenerate Cauchy numbers
Carlitz [2] has defined the degenerate Stirling numbers of the first and second kinds, ( | ) s n r λ , and S(n, r|λ) by and respectively. In a similar relation between (2) and (10) with (12), from (11) with (13) the degenerate Cauchy numbers γ n (λ) can be naturally defined in (9).
Howard [3, Theorem 3.1] determined all the coefficients of β n (λ) in (3) as In particular, the leading coefficient is the classical Cauchy numbers c n and the constant term is the classical Bernoulli numbers B n . The list of some degenerate Bernoulli numbers is in Appendix. The coefficients of γ n (λ) can also be determined. In fact, the coefficients appear in the reverse order of those of β n (λ). The list of some degenerate Cauchy numbers is in Appendix. is the Gauss hypergeometric function with the rising factorial (x) (n) = x(x + 1)⋯(x + n − 1) (n ≥ 1) and (x) (0) = 1. In this paper, (x) n = x(x − 1)⋯(x − n + 1) (n ≥ 1) is the falling factorial with (x) 0 = 1. Similar hypergeometric numbers are hypergeometric Bernoulli numbers [9][10][11] and hypergeometric Euler numbers [12,13].
Comparing the coefficients on both sides, we get the desired result. □ The hypergeometric degenerate Cauchy numbers can be expressed in terms of determinants.
Proof. The proof is obtained by induction on n. When n = 1, the result is trivial because Assume that the result is valid up to n − 1. Then, the determinant of the right-hand side (RHS) of (17) is expanded along the first row. Here, we used the recurrence relation in Proposition 1. □ The hypergeometric degenerate Cauchy numbers can be expressed explicitly. Proof. The proof is obtained by induction on n. When n = 1, the result is valid because Assume that the result is valid up to n − 1. Then, by using the recurrence relation in Proposition 1, we have There exists another form of the hypergeometric degenerate Cauchy numbers.
Proof. The proof can be obtained similar to that of Theorem 3. However, we give a different proof here. Put .

Coefficients of the hypergeometric degenerate Cauchy numbers
Some coefficients of γ N,n (λ) can also be explicitly described. In particular, the leading coefficient is the hypergeometric Bernoulli numbers B N,n and the constant is the hypergeometric Cauchy numbers c N,n . When N = 1, this result is reduced to the coefficients given in Theorem 1.
Theorem 5. The leading coefficient of γ N,n (λ) is equal to B N,n and its constant term is equal to c N,n .
Proof. From Theorem 3, we get the leading coefficient of γ N,n (λ) with λ n as Notice that Cauchy numbers are strongly related to Bernoulli numbers B n via the relation where G B (x) is the generating function of Bernoulli numbers with the rising factorial (x) (n) = x(x + 1)⋯(x + n − 1) (n ≥ 1) and (x) (0) = 1 is the confluent hypergeometric function. When N = 1, we have B n = B 1,n . One kind of generalization of Cauchy numbers is called hypergeometric Cauchy numbers c N,n ( [8]). When N = 1, we have c n = c 1,n . Nevertheless, in [16], another type of hypergeometric numbers is introduced with respect to (18). For n ≥ 0 and N ≥ 1, H-Cauchy numbers are defined by In this section, similar to H-Cauchy numbers, we shall introduce a different kind of hypergeometric degenerate Cauchy numbers with respect to (18). The H-degenerate Cauchy numbers C N,n (λ) are defined by the generating function where the hypergeometric degenerate Bernoulli numbers β N,n (γ) ( [17]) are defined by When λ → 0, the degenerate Cauchy numbers in (20) are reduced to the classical Cauchy numbers c n . We list some initial values of C N,n (λ) (0 ≤ n ≤ 5) in Appendix.
H  are the Cauchy polynomials of the first kind and those of the second kind, respectively [19,20]. Note that

Relations between two kinds of H-degenerate Cauchy polynomials
There are two kinds of H-degenerate Cauchy polynomials C N,n (λ, w; z) and C N,n (λ, −w; z). We shall show a relation between two kinds.  Remark. When λ → 0, RHS → (1 + x) (z−y−1)w . This is reduced to Theorem 6.2 in [16].
Proof. Assume that > s 0 R . Rewrite (22) with Hankel contour H 1 , where H 1 starts at 1 − e −c/λ and goes to δ > 0, and goes around the origin with radius δ, and goes back to 1 − e −c/λ . Since the integrand except for t s−1 is holomorphic in the neighborhood of t = 0, we have Then, the RHS gives the meromorphic continuation on the whole space in s. By (21), we have Z N (−n, λ, w, z) = C N,n (λ, w; z). □ The function Z N (n, λ, w, z) can be expressed as the following form.