Degenerate Analogues of Euler Zeta, Digamma, and Polygamma Functions

In recent years, much attention has been paid to the role of degenerate versions of special functions and polynomials in mathematical physics and engineering. In the present paper, we introduce a degenerate Euler zeta function, a degenerate digamma function, and a degenerate polygamma function. We present several properties, recurrence relations, infinite series, and integral representations for these functions. Furthermore, we establish identities involving hypergeometric functions in terms of degenerate digamma function.

On the matrix versions of q-zeta function, q-digamma function and q-polygamma function

This paper deals with the [Formula: see text]-analogues of generalized zeta matrix function, digamma matrix function and polygamma matrix function. We also discuss their regions of convergence, integral representations and matrix relations obeyed by them. We also give a few identities involving digamma matrix function and [Formula: see text]-hypergeometric matrix series.

Completely monotonic functions related to the q-gamma and the q-trigamma functions

In this paper, two completely monotonic functions involving the q-gamma and the q-trigamma functions where q is a positive real, are introduced and exploited to derive sharp bounds for the q-gamma function in terms of the q-trigamma function. These results, when letting q → 1, are shown to be new. Also, sharp bounds for the q-digamma function in terms of the q-tetragamma function are derived. Furthermore, an infinite class of inequalities for the q-polygamma function is established.

Sharp Inequalities for Polygamma Functions

In the paper, the authors review some inequalities and the (logarithmically) complete monotonicity concerning the gamma and polygamma functions and, more importantly, present a sharp double inequality for bounding the polygamma function by rational functions.

SPECIAL VALUES OF THE POLYGAMMA FUNCTIONS

Let q be a natural number and [Formula: see text]. We consider the Dirichlet series ∑n ≥ 1 f(n)/ns and relate its value when s is a natural number, to the special values of the polygamma function. For certain types of functions f, we evaluate the special value explicitly and use this to study linear independence over ℚ of L(k,χ) as χ ranges over Dirichlet characters mod q which have the same parity as k.