ON HURWITZ ZETA FUNCTION AND LOMMEL FUNCTIONS ATUL DIXIT AND RAHUL KUMAR Dedicated to Professor Bruce C. Berndt on the occasion of his 80th birthday Abstract. We obtain a new proof of Hurwitz's formula for the Hurwitz zeta function ζ(s; a) beginning with Hermite's formula. The aim is to reveal a nice connection between ζ(s; a) and a special case of the Lommel function Sµ,ν (z). This connection is used to rephrase a modular-type transformation involving infinite series of Hurwitz zeta function in terms of those involving Lommel functions. 1. Introduction The Hurwitz zeta function ζ(s; a) is defined for Re(s) > 1 and a 2 Cnfx 2 R : x ≤ 0g by [23, p. 36] 1 X 1 ζ(s; a) := : (n + a)s n=0 It is well-known that for ζ(s; a) can be analytically continued to the entire s-complex plane except for a simple pole at s = 1 with residue 1, and that ζ(s; 1) = ζ(s). One of the fundamental results in the theory of ζ(s; a) is the following formula of Hurwitz [23, p. 37, Equation (2.17.3)]. Theorem 1.1. For 0 < a ≤ 1 and Re(s) < 0, ( 1 1 ) 2Γ(1 − s) 1 X cos(2πna) 1 X sin(2πna) ζ(s; a) = sin πs + cos πs : (1.1) (2π)1−s 2 n1−s 2 n1−s n=1 n=1 The above result also holds1 for Re(s) < 1 if 0 < a < 1. We note that when a = 1, the above formula reduces to the functional equation of ζ(s) [23, p. 13, Equation (2.1.1)] for Re(s) < 0. which can then be seen to be true for all s 2 C by analytic continuation. Several proofs of (1.1) are available in the literature. For example, Hurwitz himself ob- tained it by transforming the Mellin transform representation of ζ(s; a) as a loop integral and then evaluating the latter. This proof can be found, for example, in [23, p. 37]. Berndt [4, Section 5] found a short proof of (1.1) by using the boundedly convergent Fourier series of 1 bxc−x+ 2 . We refer the reader interested in knowing the various proofs of this formula to [9] and the references therein (see also [10]). In [9, Section 4], Kanemitsu, Tanigawa, Tsukada 2010 Mathematics Subject Classification. Primary 11M06, 11M35; Secondary 33C10, 33C47. Keywords and phrases. Hurwitz zeta function, Lommel functions, Riemann zeta function, Hermite's formula, functional equation. 1See [1, p. 257, Theorem 12.6]. 1 2 ATUL DIXIT AND RAHUL KUMAR and Yoshimoto obtained a new proof of (1.1). Their proof commences with employing [9, Equation (4.1)] (see also [11, Equation (47)]) 1 a1−s ζ(s; a) = a−s + 2 s − 1 1 X e−2πina e2πina + Γ(1 − s; −2πina) + Γ(1 − s; 2πina) ; (−2πina)1−s (2πina)1−s n=1 which is a special case of the Ueno-Nishizawa formula [24] and then invoking the Fourier series of the Dirac-delta function δ(s). The aim of this note is to give a yet another new proof of (1.1) beginning with Hermite's well-known formula for ζ(s; a) [16, p. 609, Formula 25.11.29], valid for Re(a) > 0 and s 6= 1: 1 a1−s Z 1 sin(s tan−1(x=a)) dx ζ(s; a) = a−s + + 2 : (1.2) 2 2 s=2 2πx 2 s − 1 0 (a + x ) (e − 1) The novelty of this proof is that it reveals the connection between Hurwitz zeta function and the Lommel functions sµ,ν(z) and Sµ,ν(z) which, to the best of our knowledge, seems to have been unnoticed before. The Lommel functions are defined by [25, p. 346, equation (10)] zµ+1 1 1 3 1 1 3 1 s (z) = F 1; µ − ν + ; µ + ν + ; − z2 : (1.3) µ,ν (µ − ν + 1)(µ + ν + 1) 1 2 2 2 2 2 2 2 4 and [25, p. 347, equation (2)] µ−1 µ−ν+1 µ+ν+1 2 Γ 2 Γ 2 S (z) = s (z) + (1.4) µ,ν µ,ν sin(νπ) 1 1 × cos (µ − ν)π J (z) − cos (µ + ν)π J (z) 2 −ν 2 ν for ν2 = Z, and µ − ν + 1 µ + ν + 1 S (z) = s (z) + 2µ−1Γ Γ (1.5) µ,ν µ,ν 2 2 1 1 × sin (µ − ν)π J (z) − cos (µ − ν)π Y (z) 2 ν 2 ν for ν 2 Z, where Jν(z) and Yν(z) are Bessel functions of the first and second kinds respectively. The Lommel functions are the solutions of an inhomogeneous form of the Bessel differential equation [25, p. 345], namely, d2y dy z2 + z + (z2 − ν2)y = zµ+1: dz2 dz Lommel functions arise in mathematics, for example, in the theory of positive trigonometric sums[12]. Outside of mathematics, Lommel functions have been found to be very useful in physics as well as mathematical physics. See, for example, [2, 7, 20, 22]. Lewis [14] studied a special case of Sµ,ν(z), that is, p Cs(z) = zΓ(2s + 1)S 1 1 (z); (1.6) −2s− 2 ; 2 and represented it in terms of the incomplete gamma function. Lewis and Zagier [13] repre- sented the period functions for Maass wave forms with spectral parameter s in terms of an infinite series of Cs(z), and in the course of which they gave different representations for this special case of the Lommel function. See [13, p. 214, Proposition 1]. ON HURWITZ ZETA FUNCTION AND LOMMEL FUNCTIONS 3 In the present work, we require a new integral representation for this special case of the Lommel function Sµ,ν(z) which, to the best of our knowledge, does not seem to have been explicitly stated anywhere including [13]. This is derived in Lemma 2.1. Another ingredient needed in our proof of (1.1) is a recent result of Maˇsirevi´c[15, Theorem 2.1] (see also [3, p. 176, Theorem 5.23]) which states that for all m 2 N [ f0g; ν 2 R; x 2 1 (0; 2π) and µ > max −ν − 1; ν − 2; − 2 , 1 µ+1 X sµ,ν(kx) x 1 + µ − ν 1 + µ + ν = Γ Γ k2m+µ+1 4 2 2 k=1 (−1)mπ x2m−1 × 2Γ(m + 1 + (µ − ν)=2)Γ(m + 1 + (µ + ν)=2) 2 m ! X (−1)nζ(2m − 2n) x2n + : (1.7) Γ(n + 1 + (1 + µ − ν)=2)Γ(n + 1 + (1 + µ + ν)=2) 2 n=0 2. A new proof of Hurwitz's formula using Hermite's formula (1.2) Here we prove Theorem 1.1. To do that, however, we first need a lemma which evaluates an integral in terms of the Lommel function Sµ,ν(z). This lemma seems to be new. Lemma 2.1. Let the Lommel function Sµ,ν(z) be defined in (1.4) and (1.5). For k 2 N and a > 0, we have Z 1 −2πkx −1 e sin(s tan (x=a)) p s− 1 dx = s a(2πk) 2 S 1 1 (2πak): (2.1) 2 2 s=2 −s− ; 0 (a + x ) 2 2 Proof. We first prove the above result for Re(s) < 0 and then extend it to all s 2 C by analytic continuation. Using the inverse Mellin transform representation of the exponential function, for c1 := Re(ξ) > 0 and k > 0, Z 1 Γ(ξ) −ξ −2πky ξ y dξ = e ; (2.2) 2πi (c1) (2πk) R R d+i1 where here, and throughout the paper, (d) will always mean the integral d−i1 . Also, from [17, p. 193, Formula 5.19], for −1 < c2 := Re(ξ) < Re(s), Z −1 y 1 Γ(s − ξ)Γ(ξ) πξ ξ−s −ξ sin s tan a sin a y dξ = s : (2.3) 2 2 2 2πi (c2) Γ(s) 2 (y + a ) From (2.2), (2.3) and Parseval's identity [18, p. 82, Equation (3.1.11)] Z 1 1 Z g(x)h(x) dx = G(1 − s)H(s) ds; 0 2πi (c) where G and H are Mellin transforms of g and h respectively, for c := Re(ξ) < Re(s) and −1 < Re(ξ) < 1, Z 1 e−2πkx sin(s tan−1(x=a)) 2 Z Γ(ξ)Γ(1 − ξ)Γ(s − ξ) πξ 2 dx = sin aξ−s dξ 2 2 s=2 1−ξ 0 (a + x ) 2πi (c) (2πk) Γ(s) 2 −s Z a 1 πΓ(s − ξ) ξ−1 ξ = (2πk) a dξ Γ(s) 2πi (c) πξ cos 2 4 ATUL DIXIT AND RAHUL KUMAR a−s 1 Z 1 + ξ 1 − ξ = Γ Γ Γ(s − ξ)(2πk)ξ−1aξ dξ; Γ(s) 2πi (c) 2 2 where we used the reflection formula for the gamma function. Replacing ξ by −ξ, we see that for c := Re(ξ) > −Re(s) and −1 < Re(ξ) < 1, Z 1 e−2πkx sin(s tan−1(x=a)) a−s 1 Z 1 + ξ 1 − ξ 2 dx = Γ Γ Γ(s + ξ)(2πak)−ξ dξ: 2 2 s=2 0 (a + x ) 2πkΓ(s) 2πi (c) 2 2 (2.4) To evaluate the integral on the right-hand side of the above equation, first consider Z 1 1 + ξ 1 − ξ −ξ Is(z) := Γ Γ Γ(s + ξ)z dξ; (2.5) 2πi (c) 2 2 where c = Re(ξ).