Ewald expansions of a class of zeta-functions

The incomplete gamma function expansion for the perturbed Epstein zeta function is known as Ewald expansion. In this paper we state a special case of the main formula in Kanemitsu and Tsukada (Contributions to the theory of zeta-functions: the modular relation supremacy. World Scientific, Singapore, 2014) whose specifications will give Ewald expansions in the H-function hierarchy. An Ewald expansion for us are given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{1,2}^{2,0}\leftrightarrow H_{1,2}^{1,1}$$\end{document}H1,22,0↔H1,21,1 or its variants. We shall treat the case of zeta functions which satisfy functional equation with a single gamma factor which includes both the Riemann as well as the Hecke type of functional equations and unify them in Theorem 2. This result reveals the H-function hierarchy: the confluent hypergeometric function series entailing the Ewald expansions. Further we show that some special cases of this theorem entails various well known results, e.g., Bochner–Chandrasekharan theorem, Atkinson–Berndt theorem etc.


Background
The incomplete gamma series has a long and rich history dating back to Riemann. We begin by stating a special case of the main formula in Kanemitsu and Tsukada (2014) and state the most general modular relation which is due the third author. This is quite technical but to make the paper self-contained we have decided to include this. Then we move into stating some results involving the familiar G-function, gamma functions, confluent hypergeometric functions. In the last section which is the main section so to say, we consider the modular relation in different set up and show that these specializations entail some well known Ewald expansions. The Ewald expansions we consider are only those expansions which are described under the correspondence H 2,0 1,2 ↔ H 1,1 1,2 . k } ∞ k=1 , {β (i) k } ∞ k=1 be complex sequences. The (processing) gamma factor is with respect to the set of coefficients The (Fox) H-function is then defined by (0 ≤ n ≤ p, 0 ≤ m ≤ q, A j , B j > 0):

The modular relation
The path L is subjected to the poles separation conditions similar to the one given below. Let χ(s) be a meromorphic function which satisfies: j > 0 . For brevity we write (1) as: where the notations are obvious. The further assumptions/choices are: (1) Only finitely many poles s k (1 ≤ k ≤ L) of χ(s) are not poles of (2) The growth condition (for reals u 1 , u 2 with u 1 ≤ u ≤ u 2 ): (1) (3) L 1 (s) is a path so that the poles of lie on the right of L 1 (s), and those of lie on the left of L 1 (s). (4) L 2 (s) is chosen so that the poles of lie on the left of L 2 (s), and those of lie on the right of L 2 (s). Under these conditions the 'key-function' X(z, s | �) is defined as Then the following modular relation (Tsukada 2007) holds:

Formulas and some interesting G-function relations
In this section we recall some results involving the G-functions, gamma functions, confluent hypergeometric functions etc. We also derive/re-establish some interesting relations involving these functions.
• The confluent hypergeometric functions of the first kind is • The H → G formula: The case when C = 1 is the Meijer G-function. • The reduction-augmentation formula (Kanemitsu and Tsukada 2014, Chap. 2): • Reciprocity formula: The reciprocity formula for the gamma function and Euler's identity lead to Let us write (7) as: Then it entails G-function formula: The beta transform (or the beta integral) is a special case of Using (10) we can relate the G-functions and the confluent hypergeometric series of the first kind [also compare (Erdélyi et al. 1953, (4), p. 256) or (Prudnikov et al. 1986, 8.4.45.1, p. 715)]: In view of (17) the relation (11) reduces to and where U(a, c; z) is the confluent hypergeometric function of the second kind. One also has, (Erdélyi et al. 1953, (21), p. 266). If we take a = 1, c = 0 and change a for b then (13) reduces to Also, Thus the relation (21) in (Erdélyi et al. 1953, p. 266) becomes, Now once again using (14) with b = a relation (16) gives,

Ewald expansion for zeta-functions with a single gamma factor
We consider the general modular relation associated to Dirichlet series that satisfy the functional equation with a single gamma factor. This includes both the type of zeta-functions which satisfy the functional equation of the Riemann and that of Hecke type.
Let us consider two Dirichlet series with finite abscissas of absolute convergence σ ϕ and σ ψ , respectively. We assume that they satisfy the functional equation with a single gamma factor: or in other way,

Special cases of (19):
(1) A special case of (19) is the Dirichlet series satisfying the Riemann type functional equation with C = 1 2 , i.e.
(2) Another special case is the Dirichlet series satisfying the Hecke type of functional equation with C = 1, i.e.
with a simple pole at s = r with residue ρ. Typical examples are the Dedekind zetafunction of an imaginary quadratic field or the zeta-function associated to modular forms.
Then by Theorem 1 one has the following modular relation: Traditionally: In this form and with r = 1, the modular relation is given by the following simultaneous exchange of parameters: The following theorem which is a special case of (22) will be helpful in deducing Ewald expansions. and its special case is:

Confluent hypergeometric series
In this subsection we will deal with confluent hypergeometric series and in the next result we show it imply incomplete gamma series, i.e. Ewald expansions. The notations are all as before.
The Riemann and the Hecke type of functional equations would take the following shape in the light of the previous theorem.

(ii) The Hecke type functional equation (19) entails
Formula (28) is stated as [Kanemitsu et al. 2002, Theorem 1, (1.6)], which is a basis for the Riemann-Siegel integral formula developed in (Kanemitsu et al. 2002, §4). There are variety of specifications of (22) one of which is (24). We state one more specification which leads to the formula of Bochner and Chandrasekharan, which is of independent interest.

Theorem 5
Proof We substitute a = Cs and A = C in (22) and then apply the reduction-augmentation formula (6). That would give the proof. We obtain as an intermediate the following: Berndt (1969, Theorem 10.1) generalized the classical result of Szegö (1926) pertaining to Laguerre polynomials.

Definition 6
The Laguerre polynomial is defined by for 0 ≤ n ∈ Z. Corollary 7 (i) With C = 1 2 the Riemann type equation (19) gives (ii) With C = 1 the Hecke type functional equation (19) is (iii) Again (19) with C = 1 and 2π k in place of k entails Proof (i) Indeed, we have (ii) This is a special case of (29) with B = C = 1, b = 0 and then one needs to appeal to (3). (iii) This is the special case of Theorem 5 with B = C = 1, b = 0. We choose s = −n and appeal to the well-known relation L α n (z) = (−1) n n! �(−n, α + 1; z) = (α + 1) n n! �(−n, α + 1; z) The proof will be completed now on noting the relation between the Pochhammer symbol α n = α(α + 1) · · · (α + n − 1) and the gamma function for integer n ≥ 0. The special case of (32) with s = 0 is a formula due to Bochner (1958) [which was first proved in Bochner and Chandrasekharan (1956)] and was revisited by Berndt (1970).
Remark 8 It may be appropriate to make some comments on the characterization of the Riemann zeta-and allied functions from the functional equation of the Riemann type in reminiscence of H. Hamburger who made the first characterization of the Riemann zeta-function. The authors of Bochner and Chandrasekharan (1956), Chandrasekharan and Mandelbrojt (1959) and Chandrasekharan and Mandelbrojt (1957) are concerned with bounding the number of linearly independent solutions to the functional Eq. (19), thus leading to the uniqueness of the solution. In all these investigations a special case of (31) plays an essential role, which in turn was suggested by Siegel's simplest proof Siegel (1922) of Hamburger's theorem. Here one sees the importance of exhausting those relations that are equivalent to the functional equation.
We shall use Theorem 5 to show that (next corollary) the Atkinson-Berndt Abel mean is nothing but another way to prove the functional equation.