EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1747-1761 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some Explicit Formulas of Hurwitz Lerch type Poly-Cauchy Polynomials and Poly-Bernoulli Polynomials Noel B. Lacpao Department of Mathematics, College of Arts and Sciences, Bukidnon State University, Malaybalay City, 8700, Philippines Abstract. In this paper, the Hurwitz-Lerch poly-Cauchy and poly-Bernoulli polynomials are de- fined using polylogarithm factorial function. Some properties of these types of polynomials were also established. Specifically, two different forms of explicit formula of Hurwitz-Lerch type poly- Cauchy polynomials were obtained using Stirling numbers of the first and second kind and an explicit formula of Hurwitz-Lerch type poly-Bernoulli polynomials was established using the Stir- ling numbers of the first kind. 2020 Mathematics Subject Classifications: 11M35, 11B83, 11B68, 11B73, 05A19 Key Words and Phrases: Polylogarithm factorial functions, poly-Cauchy numbers of the first and second kind, poly-Bernoulli numbers, Hurwitz–Lerch factorial zeta function, generating func- tion 1. Introduction It is known that Euler’s constant appeared many times in different well-known ex- pressions or formulas such as in exponential integral, the Laplace transform of the natural logarithm, the first of the Laurent series expansion for the Riemann Zeta function, solution of the second kind to Bessel’s equation and many more. Surprisingly, the Cauchy num- bers have appeared in the formula involving Euler’s constant [16]. This fact has attracted several researchers to work further on Cauchy numbers. The Cauchy numbers [6, 12, 15] of the first and second kind, respectively denoted by cn and ĉn, are usually defined by its generating functions: t ln(1 + t) = ∞∑ n=0 cn tn n! , (|t| < 1) DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4825 Email address: noel.lacpao@buksu.edu.ph (N. Lacpao) https://www.ejpam.com 1747 © 2023 EJPAM All rights reserved. N. B. Lacpao / Eur. J. Pure Appl. Math, 16 (3) (2023), 1747-1761 1748 and t (1 + t) ln(1 + t) = ∞∑ n=0 ĉn tn n! , (|t| < 1). The Bernoulli numbers [1] denoted by Bn are defined by the generating function t et − 1 = ∞∑ n=0 Bn tn n! , (|t| < 2π). One of its combinatorial relations is given by Bn = (−1)n n∑ m=0 [ n m ] (−1)mm! m+ 1 . The Cauchy numbers appear in the Laplace Summation Formula [15] as a coefficient and are also called the Cauchy numbers of the first kind. In this formula, the Cauchy numbers are expressed in terms of Stirling numbers as follows,∫ f(t)dt = ∆−1 ∞∑ k=0 ck k! ∆k where ∆ is the forward difference operator. This is analogous to Euler McLaurin Summa- tion Formula where Bernoulli numbers are expressed in terms of Stirling numbers, however, differentiation is being used instead of the difference operators as shown below: b−1∑ k=a f(k) = ∫ b a f(x)dx+ n∑ v=1 v! Bv f (v−1)(x) ∣∣∣b a −Rn[f ]. A variation of Cauchy numbers of the first kind was introduced by Komatsu [12] inspired by the polylogarithm factorial functions Lifk(z) = ∞∑ m=0 zm m!(m+ 1)k . These numbers are called poly-Cauchy numbers of the first and second kind, denoted by c (k) n and ĉ (k) n , respectively. More precisely, these numbers are defined by means of integrals as follows: c(k)n = n! ∫ 1 0 · · · ∫ 1 0 ( t1t2 · · · tk n ) dt1dt2 · · · dtk and ĉ(k)n = n! ∫ 1 0 · · · ∫ 1 0 ( −t1t2 · · · tk n ) dt1dt2 · · · dtk. These numbers have combinatorial relations with Stirling numbers of the first and second kind as follows N. B. Lacpao / Eur. J. Pure Appl. Math, 16 (3) (2023), 1747-1761 1749 n∑ m=0 { n m } c(k)m = 1 (n+ 1)k , n∑ m=0 { n m } ĉ(k)m = (−1)n (n+ 1)k and explicit formulas c(k)n = (−1)n n∑ m=0 [ n m ] (−1)m (m+ 1)k , ĉ(k)n = (−1)n n∑ m=0 [ n m ] 1 (m+ 1)k where [ n m ] and { n m } are the Stirling numbers of the first and second kind, respectively, with generating functions: [ln(1 + t)]m m! = ∞∑ n=m (−1)n−m [ n m ] tn n! , (|t| < 1) and (et − 1)m m! = ∞∑ n=m { n m } tn n! , (|t| < 1) such that [ n m ] = 0 and { n m } = 0 for n < m. Parallel to this, Kaneko [11] defined certain variation of Bernoulli numbers in terms of polylogarithm function Lik(z) = ∞∑ n=1 zn nk , (|z| < 1) which are called poly-Bernoulli numbers denoted by B (k) n . These types of numbers are defined by Lik(1− e−t) 1− e−t = ∞∑ n=0 B(k) n tn n! . Certain generalization of poly-Cauchy numbers of the first and second kind was in- troduced by Cenkci and Young [4]. This generalization was motivated by the concept of Hurwitz-Lerch factorial zeta function defined by Φf(z, s, a) = ∞∑ n=0 zn n!(n+ a)s for s ∈ C when |z| < 1, Re s > 1 when |z| = 1 and a /∈ {0,−1,−2, · · · }. These numbers were called Hurwitz type poly-Cauchy numbers of the first and second kind, denoted by N. B. Lacpao / Eur. J. Pure Appl. Math, 16 (3) (2023), 1747-1761 1750 c (k) n (a) and ĉ (k) n (a), which are respectively defined by Φf(log(1 + t), k, a) = ∞∑ n=0 c(k)n (a) tn n! and Φf(− log(1 + t), k, a) = ∞∑ n=0 ĉ(k)n (a) tn n! . These numbers possessed the following properties which are analogous to those of poly- Cauchy numbers: explicit formulas c(k)n (a) = (−1)n n∑ m=0 (−1)mS1(n,m) (m+ a)k , ĉ(k)n (a) = (−1)n n∑ m=0 S1(n,m) (m+ a)k , relations with Stirling numbers of the second kind n∑ m=0 S2(n,m)c(k)m (a) = 1 (n+ a)k , n∑ m=0 S2(n,m)ĉ(k)m (a) = (−1)n (n+ a)k , and expressions of Hurwitz type poly-Bernoulli numbers in terms of Hurwitz type poly- Cauchy numbers B(k) n (a) = n∑ l=0 n∑ m=0 (−1)m+nm!S2(n,m)S2(m, l)c (k) l (a), B(k) n (a) = n∑ l=0 n∑ m=0 (−1)mm!S2(n,m)S2(m, l)ĉ (k) l (a), c(k)n (a) = n∑ l=0 n∑ m=0 (−1)m+n m! S1(n,m)S1(m, l)B (k) l (a), ĉ(k)n (a) = n∑ l=0 n∑ m=0 (−1)n m! S1(n,m)S1(m, l)B (k) l (a). Recently, several generalizations of these numbers have been introduced relating to some well-known special numbers. For instance, the poly-Cauchy polynomials are ex- pressed in terms of polylogarithm factorial function and multi poly-Cauchy polynomials, multi poly-Bernoulli and multi poly-Euler numbers and polynomials are expressed in terms N. B. Lacpao / Eur. J. Pure Appl. Math, 16 (3) (2023), 1747-1761 1751 of multiple polylogarithm factorial function [7, 8, 10]. Moreover, other well known families of polynomials such as the Apell-type classical polynomials and Apostol-type polynomi- als have attracted research attention due to their important applications in the areas of applied mathematics, physics and engineering [3, 5]. This present study aims to establish other variation of generalizing Cauchy and Bernoulli polynomials that can be related to the well-known Hurwitz-Lerch factorial zeta function. The generalization may contribute to the development of numerous applications in number theory, numerical analysis and difference-differential equations. 2. Hurwitz-Lerch type Poly-Cauchy and Poly-Bernoulli Polynomials Kamano and Komatsu [13] defined the poly-Cauchy polynomials of the first and second kind, c (k) n (x) and ĉ (k) n (x), respectively, as follows: c(k)n (x) = n! ∫ 1 0 · · · ∫ 1 0 ( t1t2 · · · tk + x n ) dt1dt2 · · · dtk, (k ≥ 1) and ĉ(k)n (x) = n! ∫ 1 0 · · · ∫ 1 0 ( −t1t2 · · · tk − x n ) dt1dt2 · · · dtk, (k ≥ 1) with generating functions (1 + t)xLifk(ln(1 + t)) = ∞∑ n=0 c(k)n (x) tn n! (1) Lifk(− ln(1 + t)) (1 + t)x = ∞∑ n=0 ĉ(k)n (x) tn n! (2) where Lifk(z) = ∞∑ m=0 zm m!(m+ 1)k . (3) Observe that from (3), we get Φf(ln(1 + t), k, a) = ∞∑ n=0 (ln(1 + t))n n!(n+ a)k = Lifk(ln(1 + t))(a). If a = 1, we get Φf(z, k, 1) = Lifk(z). Comparing this with the left hand side of (1) and (2), it would be logical to define the Hurwitz-Lerch poly-Cauchy polynomials of the first and seconds kind as follows: Definition 1. The Hurwitz-Lerch type poly-Cauchy polynomials of the first kind denoted by c (k) n,a(x) are defined by (1 + t)xΦf(ln(1 + t), k, a) = ∞∑ n=0 c(k)n,a(x) tn n! . N. B. Lacpao / Eur. J. Pure Appl. Math, 16 (3) (2023), 1747-1761 1752 Definition 2. The Hurwitz-Lerch type poly-Cauchy polynomials of the second kind denoted by ĉ (k) n,a(x) are defined by Φf(− ln(1 + t), k, a) (1 + t)x = ∞∑ n=0 ĉ(k)n,a(x) tn n! . These polynomials have an explicit formula involving the Stirling numbers of the first and second kind. Theorem 1. For k ∈ Z, n ≥ 0 we have c(k)n,a(x) = n∑ s=0 x! (x− n+ s)! (−1)s−m ( n s ) s∑ m=0 [ s m ] (m+ a)k . Proof. ∞∑ n=0 c(k)n,a(x) tn n! = (1 + t)xΦf(ln(1 + t), k, a). Working on the right hand side, we have (1 + t)xΦf(ln(1 + t), k, a) = ( ∞∑ s=0 ( x s ) ts )( ∞∑ m=0 (ln(1 + t))m m!(m+ a)k ) = ( ∞∑ s=0 ( x s ) ts )( ∞∑ m=0 ( ∞∑ n=m (−1)n−m [ n m ] (m+ a)k )) = ∞∑ s=0 s∑ n=0 {( x s− n ) (−1)n−mts−n ( n∑ m=0 ( [ n m ] (m+ a)k tn n! ))} = ∞∑ s=0 { s∑ n=0 x!(−1)n−m (s− n)!(x− s+ n)! ( n∑ m=0 ( [ n m ] (m+ a)k tss! n!s! ))} = ∞∑ s=0 { s∑ n=0 x! (x+ s− n)! (−1)n−m ( n∑ m=0 ( [ n m ] (m+ a)k ) s! n!(s− n)! ts s! )} = ∞∑ s=0 { s∑ n=0 x! (x− s+ n)! (−1)n−m ( s n )( n∑ m=0 ( [ n m ] (m+ a)k ))} ts s! = ∞∑ n=0 { n∑ s=0 x! (x− n+ s)! (−1)s−m ( n s )( s∑ m=0 ( [ s m ] (m+ a)k ))} tn n! . Comparing the coefficients completes the proof. ■ Theorem 2. For k ∈ Z, n ≥ 0 we have ĉ(k)n,a(x) = n∑ s=0 (x+ n− s− 1)! (x− 1)! (−1)n ( n s ) s∑ m=0 [ s m ] (m+ a)k . N. B. Lacpao / Eur. J. Pure Appl. Math, 16 (3) (2023), 1747-1761 1753 Proof. ∞∑ n=0 ĉ(k)n,a(x) tn n! = 1 (1 + t)x Φf(− ln(1 + t), k, a). Working on the right hand side, we have 1 (1 + t)x Φf(− ln(1 + t), k, a) = ( ∞∑ s=0 ( x+ s− 1 s ) (−1)sts )( ∞∑ m=0 (− ln(1 + t))m m!(m+ a)k ) = ( ∞∑ s=0 ( x+ s− 1 s ) (−1)sts )( ∞∑ m=0 ( (ln(1 + t))m(−1)m m! 1 (m+ a)k ) = ( ∞∑ s=0 ( x+ s− 1 s ) (−1)sts )( ∞∑ m=0 ( ∞∑ n=m (−1)n [ n m ] (m+ a)k ) tn n! ) = ∞∑ s=0 s∑ n=0 {( x+ s− n− 1 s− n ) (−1)sts−n ( n∑ m=0 ( [ n m ] (m+ a)k ) tn n! )} = ∞∑ s=0 { s∑ n=0 (x+ s− n− 1)! (s− n)!(x− 1)! (−1)s ( n∑ m=0 ( [ n m ] (m+ a)k ) tss! n!s! )} = ∞∑ s=0 { s∑ n=0 (x+ s− n− 1)! (x− 1)! (−1)s ( n∑ m=0 ( [ n m ] (m+ a)k ) s! n!(s− n)! ts s! )} = ∞∑ s=0 { s∑ n=0 (x+ s− n− 1)! (x− 1)! (−1)s ( s n )( n∑ m=0 ( [ n m ] (m+ a)k ))} ts s! = ∞∑ n=0 { n∑ s=0 (x+ n− s− 1)! (x− 1)! (−1)n ( n s )( s∑ m=0 ( [ s m ] (m+ a)k ))} tn n! . Comparing the coefficients yields the result. ■ The Hurwitz-Lerch type poly-Bernoulli Polynomials can also be defined by means of Hurwitz-Lerch zeta function Φ(z, s, a). Definition 3. The Hurwitz-Lerch type poly-Bernoulli polynomials denoted by B (k) n,a(x) are defined by Φ(1− e−t, k, a)etx = ∞∑ n=0 B(k) n,a(x) tn n! . where etx = ∞∑ n=0 (tx)n n! . Bayad and Hamahata [2] established an explicit formula for poly-Bernoulli polynomi- als. Analogous to this, the Hurwitz-Lerch type poly-Bernoulli polynomials have explicit explicit formula involving Stirling numbers. N. B. Lacpao / Eur. J. Pure Appl. Math, 16 (3) (2023), 1747-1761 1754 Theorem 3. For k ∈ Z, n ≥ 0 we have B(k) n,a(x) = n∑ s=0 xn−s(−1)s−m ( n s ) s∑ m=0 { s m } m! (m+ a)k . Proof. ∞∑ n=0 B(k) n,a(x) tn n! = Φ(1− e−t, k, a)ext. Working on the right hand side, we have extΦ(1− e−t, k, a) = ( ∞∑ s=0 (xt)s s! )( ∞∑ m=0 (1− e−t)mr (m+ a)k ) = ( ∞∑ s=0 (xt)s s! )( ∞∑ m=0 (e−t − 1))m(−1)m m! m! (m+ a)k ) = ( ∞∑ s=0 (xt)s s! )( ∞∑ m=0 ( ∞∑ n=m (−1)n−m { n m } m! (m+ a)k ) tn n! ) = ∞∑ s=0 s∑ n=0 { xs−n (s− n)! (−1)n−mts−n ( n∑ m=0 ( { n m } m! (m+ a)k ) tn n! )} = ∞∑ s=0 s∑ n=0 { xs−n(−1)n−m ( n∑ m=0 ( { n m } m! (m+ a)k ) s! (s− n)!n! ts s! )} = ∞∑ s=0 { s∑ n=0 xs−n(−1)n−m ( s n )( n∑ m=0 ( { n m } m! (m+ a)k ))} ts s! = ∞∑ n=0 { n∑ s=0 xn−s(−1)s−m ( n s )( s∑ m=0 ( { s m } m! (m+ a)k ))} tn n! Comparing the coefficients gives the result. ■ 3. Hurwitz-Lerch type Multi-Poly-Cauchy and Multi-Poly-Bernoulli Polynomials Further generalization of poly-Cauchy numbers of the first and second kind was defined by Lacpao et al. [14] in polynomial form by means of multiple polylogarithm function Lik1,k2,··· ,kr(z) = ∑ 0