Haruo Tsukada’s research while affiliated with Kindai University and other places

To Professor Hidenori Fujiwara on the occasion of his 65th birthday, with friendship and respect Abstract. The importance of the Hecke gamma-transform in number theory cannot be overstated. Along with this, there has been equally eective applications of the method of beta-transform, or the binomial expansion, leading to the K-Bessel series. The purpose here is to elucidate the explicit and implicit use of the beta-transform in the transformation of the perturbed Dirichlet series and reveal the hidden structure as the Fourier-Bessel expansion. In the rst instance, we shall deal with Stark's method along with a recent result of Murty-Sinha as a manifestation of the beta-transform. Later and in the main part, we shall resurrect the long-forgotten important work of Koshlyakov [10, 11, 12, 13] showing that Koshlyakov's formula for the perturbed Dedekind zeta-functions for a real quadratic eld is in fact Lipschitz summation formula, and that for an imaginary quadratic eld the K-Bessel function is intrinsic to the Hecke functional equation. Similarly, we shall elucidate Koshlyalov's σ-series in terms of Kelvin functions.

The importance of the Hecke gamma-transform in number theory cannot be overstated. Along with this, there has been equally effective applications of the method of beta-transform, or the binomial expansion, leading to the K-Bessel series. The purpose here is to elucidate the explicit and implicit use of the beta-transform in the transformation of the perturbed Dirichlet series and reveal the hidden structure as the Fourier- Bessel expansion. In the rst instance, we shall deal with Stark's method along with a recent result of Murty-Sinha as a manifestation of the betatransform. Later and in the main part, we shall resurrect the long-forgotten important work of Koshlyakov showing that Koshlyakov's formula for the perturbed Dedekind zeta-functions for a real quadratic eld is in fact Lipschitz summation formula, and that for an imaginary quadratic eld the K-Bessel function is intrinsic to the Hecke functional equation. Similarly, we shall elucidate Koshlyalov's �-series in terms of Kelvin functions

The incomplete gamma function expansion for the perturbed Epstein zeta function is known as Ewald expansion in physics and in chemistry. In this paper we state a special case of the main formula by Haruo Tsukada (giving the modular relation) whose specifications will give almost all existing Ewald expansions in the H-function hierarchy. An Ewald expansion for us are those formulas given by $H_{1,2}^{2,0}\leftrightarrow H_{1,2}^{1,1}$ or its variants and especially the incomplete gamma expansion. We shall treat the case of a single gamma factor which includes both the Riemann as well as the Hecke type of functional equations and unify them in the main theorem ( theorem 2.1). This result reveals the H-function hierarchy: the confluent hypergeometric function series entailing the Ewald expansions. Further some special cases of this theorem entails various well known results, e.g., Bochner-Chandrasekharan theorem, Atkinson-Berndt theorem etc.

We consider the zeta functions satisfying the functional equation with multiple gamma factors and prove a far-reaching theorem,an intermediate modular relation, which gives rise to many (including many of the hitherto found) arithmetical Fourier series as a consequence of the functional equation. Typical examples are the Diophantine Fourier series considered by Hardy and Littlewood and one considered by Hartman and Wintner, which are reciprocals of each other, in addition to our previous work. These have been thoroughly studied by Li, Ma and Zhang. Our main contribution is to the effect that the modular relation gives rise to the Fourier series for the periodic Bernoulli polynomials and Kummer’s Fourier series for the log sin function, thus giving a foundation for a possible theory of arithmetical Fourier series based on the functional equation.

In this paper, we continue our previous investigations on applications of the Epstein zeta-functions. We shall mostly state the results for the lattice zeta-functions, which can be immediately translated into those for the corresponding Epstein zeta-functions. We shall take up the generalized Chowla-Selberg (integral) formula and state many concrete special cases of this formula.

This book (Vista II), is a sequel to Vistas of Special Functions (World Scientific, 2007), in which the authors made a unification of several formulas scattered around the relevant literature under the guiding principle of viewing them as manifestations of the functional equations of associated zeta-functions. In Vista II, which maintains the spirit of the theory of special functions through zeta-functions, the authors base their theory on a theorem which gives some arithmetical Fourier series as intermediate modular relations — avatars of the functional equations. Vista II gives an organic and elucidating presentation of the situations where special functions can be effectively used. Vista II will provide the reader ample opportunity to find suitable formulas and the means to apply them to practical problems for actual research. It can even be used during tutorials for paper writing. © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.

his book (Vista II), is a sequel to Vistas of Special Functions (World Scientific, 2007), in which the authors made a unification of several formulas scattered around the relevant literature under the guiding principle of viewing them as manifestations of the functional equations of associated zeta-functions. In Vista II, which maintains the spirit of the theory of special functions through zeta-functions, the authors base their theory on a theorem which gives some arithmetical Fourier series as intermediate modular relations — avatars of the functional equations. Vista II gives an organic and elucidating presentation of the situations where special functions can be effectively used. Vista II will provide the reader ample opportunity to find suitable formulas and the means to apply them to practical problems for actual research. It can even be used during tutorials for paper writing.

We shall locate Katsurada’s results, in our framework of modular relations, on two series involving the values of the Riemann zeta-function, which are decisive generalizations of earleir results of Chowla and Hawkins and of Buschman and Srivastava \textit{et al.} We shall elucidate these results as an improper or a proper modular relation according as the involved parameter $\nu$ exerts effects on the series or not, eventually indicating that they are disguised form of modular relations as given by Theorem 4 in 3.