Modular relations involving generalized digamma functions

Generalized digamma functions ψ<inf>k</inf>(x), studied by Ramanujan, Deninger, Dilcher, Kanemitsu, Ishibashi etc., appear as the Laurent series coefficients of the Hurwitz zeta function. In this paper, a modular relation of the form F<inf>k</inf>(α)=F<inf>k</inf>(1/α) containing infinite series of ψ<inf>k</inf>(x), or, equivalently, between the generalized Stieltjes constants γ<inf>k</inf>(x), is obtained for any k∈N. When k=0, it reduces to a famous transformation given on page 220 of Ramanujan's Lost Notebook. For k=1, an integral containing Riemann's Ξ-function, and corresponding to the aforementioned modular relation, is also obtained along with its asymptotic expansions as α→0 and α→∞. Carlitz-type and Guinand-type finite modular relations involving ψ<inf>j</inf>^(m)(x),0≤j≤k,m∈N∪{0}, are also derived, thereby extending previous results on the digamma function ψ(x). The extension of Guinand's result for ψ<inf>j</inf>^(m)(x),m≥2, involves an interesting combinatorial sum h(r) over integer partitions of 2r into exactly r parts. This sum plays a crucial role in an inversion formula needed for this extension. This formula has connection with the inversion formula for the inverse of a triangular Toeplitz matrix. The modular relation for ψ<inf>j</inf>^′(x) is subtle and requires delicate analysis.