Modular relations involving generalized digamma functions

Abstract

Generalized digamma functions $ψ_k(x)$, studied by Ramanujan, Deninger, Dilcher, Kanemitsu, Ishibashi etc., appear as the Laurent series coefficients of the zeta function associated to an indefinite quadratic form. In this paper, a modular relation of the form $F_k(α)=F_k(1/α)$ containing infinite series of $ψ_k(x)$, or, equivalently, between the generalized Stieltjes constants $γ_k(x)$, is obtained for any $k\in\mathbb{N}$. 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 $α\to0$ and $α\to\infty$. Carlitz-type and Guinand-type finite modular relations involving $ψ_j^{(m)}(x), 0\leq j\leq k, m\in\mathbb{N}\cup\{0\},$ are also derived, thereby extending previous results on the digamma function $ψ(x)$. The extension of Guinand's result for $ψ_j^{(m)}(x), m\geq2,$ 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 $ψ_j'(x)$ is subtle and requires delicate analysis.