The Lerch zeta and related functions of non-positive integer order.

It is known that the Lerch (or periodic) zeta function of non-positive integer order, $ell _{-n}(xi )$, $nin mathbb {N}_{0}:= {0, 1, 2, 3, ldots }$, is a polynomial in $cot (pi xi )$ of degree $n + 1$. In this paper, a very simple explicit closed-form formula for this polynomial valid for any degree is derived. In addition, novel analogous explicit closed-form formulae for the Legendre chi function, the alternating Lerch zeta function and the alternating Legendre chi function are established. The obtained formulae involve the Carlitz-Scoville tangent and secant numbers of higher order, and the derivative polynomials for tangent and secant are used in their derivation. Several special cases and consequences are pointed out, and some examples are also given. [ABSTRACT FROM AUTHOR]