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A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function

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A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function mathematics Article A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function Daeyeoul Kim 1,* and Yilmaz Simsek 2 1 Department of Mathematics and Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju 54896, Korea 2 Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya TR-07058, Turkey; [email protected] * Correspondence: [email protected] Abstract: In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol–Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials. Keywords: Bernoulli numbers and polynomials; Euler numbers and polynomials; Apostol–Bernoulli and Apostol–Euler numbers and polynomials; Hurwitz–Lerch zeta function; Hurwitz zeta function; alternating Hurwitz zeta function; generating function; Mellin transformation MSC: 05A15; 11B68; 26C0; 11M35 Citation: Kim, D.; Simsek, Y. A New Family of Zeta Type Function 1. Introduction Involving the Hurwitz Zeta Function The families of zeta functions and special numbers and polynomials have been studied and the Alternating Hurwitz Zeta widely in many areas. They have also been used to model real-world problems. In addition Function. Mathematics 2021, 9, 233. to these functions, generating functions for special numbers and polynomials have also https://doi.org/10.3390/math9030233 been used commonly in variety of studies (cf. [1–34]). The motivation of this paper is to apply the Mellin transformation to the following Received: 14 December 2020 generating function: Accepted: 23 January 2021 Published: 25 January 2021 ¥ wn aw F (w, k, a) = Y (k, a) = , (1) Y ∑ n (k+1)w −(k+1)w w −w n= n! e − e + e − e Publisher’s Note: MDPI stays neu- 0 tral with regard to jurisdictional clai- where k 2 Z and a, and w 2 R (or C) (cf. [25]). This yields a new family of zeta type ms in published maps and institutio- functions involving the Hurwitz zeta function, the alternating Hurwitz zeta function, nal affiliations. and the Lerch zeta function. After investigating some properties of these functions, we found that these functions can interpolate the Apostol–Bernoulli polynomials, the Bernoulli numbers and polynomials, and the Euler numbers and polynomials at negative integers. Copyright: © 2021 by the authors. Li- The following notations, definitions and relations are used in this paper: Let N, Z, Q, censee MDPI, Basel, Switzerland. R and C denote the set of natural numbers, the set of integers, the set of rational numbers, This article is an open access article the set of real numbers and the set of complex numbers, respectively. − − − distributed under the terms and con- N0= N [ f0g and Z0 := Z [ f0g, Z := f−1, −2, −3, · · · g. ditions of the Creative Commons At- Moreover, the falling factorial is given by: tribution (CC BY) license (https:// creativecommons.org/licenses/by/ x(x − 1)(x − 2) ... (x − n + 1) if n 2 N, (x)n = 4.0/). 1 if n = 0, Mathematics 2021, 9, 233. https://doi.org/10.3390/math9030233 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 233 2 of 11 where x 2 R. The Apostol–Bernoulli polynomials Bn(y; l) are defined by means of the following generating function: wewy ¥ wn F (w, y; l) = = B (y; l) , (2) A w − ∑ n le 1 n=0 n! (cf. [1]). Substituting y = 0 into (2), we have: B0(0; l) = 0 and lB1(1; l) = 1 + B1(l) and for n ≥ 2, lBn(1; l) = Bn(l), (cf. [1]). By using (2), we have: n n n−j Bn(y; l) = ∑ y Bj(l) (3) j=0 j (cf. [1]). By using (3), the first few values of the Apostol–Bernoulli polynomials are given as follows: 1 (l − 1)y − 2l B1(y; l) = , B2(y; l) = , l − 1 (l − 1)2 3(l − 1)2y2 − 6l(l − 1)y + 3l(l + 1) B3(y; l) = , (l − 1)3 and so on. Substituting y = 1 into (2), we have the following Apostol–Bernoulli numbers: n−1 n Bn(1, l) − Bn(l) = ∑ Bj(l), j=0 j where Bn(l) = Bn(0; l). Using (2), one has the following well-known identity: m−1 m (1 − l)Bm(l) − l ∑ Bj(l) = 0 (4) j=0 j (cf. [24,29,31]). By using the above identity, we have the following few values of the Apostol–Bernoulli numbers: 1 −2l 3l(l + 1) −4l l2 + 4l + 1 B1(l) = , B2(l) = , B3(l) = , B4(l) = , l − 1 (l − 1)2 (l − 1)3 (l − 1)4 and so on (cf. [1,13,15,17–34]). Mathematics 2021, 9, 233 3 of 11 The Bernoulli polynomials Bn(y) are defined by means of the following generating function: wewy ¥ wn F (w, y) = = B (y) (5) B w − ∑ n e 1 n=0 n! (cf. [1,4–6,8,9,13–34]). Hence, a few values of the Bernoulli polynomials are given as follows: x − 1 x2 − x + 1 2x3 − 3x2 + 1x B (x) = 1, B (x) = , B (x) = , B (x) = , 0 1 2 2 6 3 2 30x4 − 60x3 + 30x2 − 1 6x5 − 15x4 + 10x3 − x B (x) = , B (x) = , 4 30 5 6 and so on. Since Bn = Bn(0) denotes the Bernoulli numbers, a few values of these numbers are given as follows: 1 1 1 1 B = 1, B = − , B = , B = 0, B = − , B = 0, B = , B = 0, 0 1 2 2 6 3 4 30 5 6 42 7 and so on, here note that B2n+1 = 0 for n ≥ 2 (cf. [1,4–6,8–10,13–32]). The Apostol–Euler polynomials of the first kind En(x, l) are defined by means of the following generating function: 2 ¥ tn F (t, x; k, l) = etx = E (x, l) , (6) P1 t + ∑ n le 1 n=0 n! and by using (6), we have: n n n−j En(x; l) = ∑ x Ej(l) (7) j=0 j (cf. [13,15,17–34]). By combining (6) with (2), we have the following well-known relation: (n + 1)En(x; l) + 2Bn+1(x; −l) = 0 (8) (cf. [29,30]). Substituting l = 1 into (6), we have the Euler polynomials of the first kind; that is: En(x) = En(x; 1). With the help of the Equation (7), a few values of the Euler polynomials of the first kind are given as follows: 2x − 1 4x3 − 6x2 + 1 E (x) = 1, E (x) = , E (x) = x2 − x, E (x) = , 0 1 2 2 3 4 2x5 − 5x4 + 5x2 − 1 E (x) = x4 − 2x3 + x, E (x) = , 4 5 2 and so on (cf. [1,4–6,8,9,13–34]). Putting x = 0 in (6), we have the Apostol–Euler numbers: En(l) = En(0, l), where 2 E (l) = . 0 l + 1 Mathematics 2021, 9, 233 4 of 11 By using (6), one has the following well-known identity: l m−1 m E (l) = − E (l) (9) m + ∑ j j 1 l j=0 (cf. [24,29,31]). By using the above identity, we have the following few values of the Apostol–Euler numbers: 2l 2l(l − 1) 2l l2 − 4l + 1 E1(l) = − , E2(l) = , E3(l) = − , (l + 1)2 (l + 1)3 (l + 1)4 and so on. Setting l = 1 into (6), we have the Euler numbers of the first kind: (1) En = En (1) = En(0), which are given by: 2 ¥ tn = E t + ∑ n e 1 n=0 n! (cf. [1,4–6,8,9,13–34]). Hence a few of the values of the Euler numbers of the first kind are given as follows: 1 1 1 17 31 E = 1, E = − , E = 0, E = , E = − , E = , E = − , 0 1 2 2 3 4 5 2 7 8 9 2 and so on, here noting that E2n = 0 for n ≥ 1 (cf. [1,4–6,8–11,13–34]). ∗ The Euler numbers of the second kind Ej are given by means of the following gener- ating function: 2 ¥ tj = E∗ . t + −t ∑ j e e j=0 j! ∗ We now give the following well-known relation between the numbers En and En: 1 E∗ = 2nE n n 2 n n j = ∑ 2 Ej. j=0 j Using the above equation, we have: ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ E0 = 1, E2 = −1, E3 = 0, E4 = 5, E5 = 0, E6 = −61, E7 = 0, E8 = 1385, ∗ and so on, and here noting that E2n+1 = 0 for n ≥ 0 (cf. [1,4–6,8–10,13–34]). ∗ In [21], Simsek gave many remarks and observations on the numbers En and En. Simsek [21] divided the Euler numbers into two groups, naming them as the Euler numbers ∗ of the first kind En and the Euler numbers of the second kind En. The alternating Hurwitz zeta function can be used to interpolate these numbers at negative integers. By using Equation (1), we have: awe(k+1)w FY (w, k, a) = , (10) e(k+2)w − 1ekw + 1 (cf. [25]). Mathematics 2021, 9, 233 5 of 11 Equation (10) yields the following relations among the numbers Yn(k, a), the Bernoulli polynomials and numbers, and the Euler polynomials and numbers: n n−1 1 1 n Yn(0, 2) = 2 Bn = ∑ Bj(1)En−j, 2 2 j=0 j n n n−j−1 2 Yn(1, 2) = ∑ 3 Bn−j Ej, j=0 j 3 and n n n−j−1 Yn(1, 2) = ∑ 3 Bn−jEj(2), j=0 j (cf.
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