We show that if $P(x)$ is a polynomial with nondecreasing, nonnegative coefficients, then the coefficient sequence of $P(x+1)$ is unimodal. Applications are given.

We present a rational version of the classical Landen transformation for elliptic integrals. This is employed to obtain explicit closed-form expressions for a large class of integrals of even rational functions and to develop an algorithm for numerical integration of these functions.

Let be a sequence that satisfies a first order homogeneous recurrence <!-- MATH $t_n = Q(n)t_{n-1}$ --> , where is a polynomial with integer coefficients. We describe the asymptotic behavior of the -adic valuation of .

We study the properties of a function ψ(z, q) (the generalized polygamma function), intimately connected with the Hurwitz zeta function and defined for complex values of the variables z and q, which is entire in the variable z and reduces to the usual polygamma function ψ^(m)(q) for z a non-negative integer m, and to the balanced negapolygamma function ψ^(−m)(q) introduced in Ref. [5] Espinosa, O. and Moll, V. (2002). On some integrals involving the Hurwitz zeta function: Part 2. The Ramanujan Journal, 6: 449–468. [Google Scholar] for z a negative integer −m.