Related papers: Explicit asymptotics for certain single and double…
This paper argues that automated proofs of identities for non-terminating hypergeometric series are feasible by a combination of Zeilberger's algorithm and asymptotic estimates. For two analogues of Saalsch\"utz' summation formula in the…
Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{\nu=0}^{\infty} a_{\nu}$ of infinite series $\sum_{\nu=0}^{\infty} a_{\nu}$ for special functions are constructed by solving a system of linear equations.…
We introduce finite and symmetric Mordell-Tornheim type of multiple zeta values and give a new approach to the Kaneko-Zagier conjecture stating that the finite and symmetric multiple zeta values satisfy the same relations.
Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new…
In the present paper, employing properties of the complete elliptic integrals of the first and second kind, we deduce closed-form formulae for the lattice sums and other new formulae. Applications to the effective properties of regular and…
The cotangent zeta function is a very interesting object, which is related to partial zeta functions and Hecke $L$-functions of real quadratic fields. Its special values at odd integers greater than 1 are explicitly evaluated by Berndt in…
Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then,…
Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle…
We present highlights of computations of the Riemann zeta function around large values and high zeros. The main new ingredient in these computations is an implementation of the second author's fast algorithm for numerically evaluating…
In this paper, we establish some expressions of Mneimneh-type binomial sums involving multiple harmonic-type sums in terms of finite sums of Stirling numbers, Bell numbers and some related variables. In particular, we present some new…
We consider integrals on unitary groups $U_d$ of the form $$\int_{U_d}U_{i_1j_1}... U_{i_qj_q}U^*_{j'_{1}i'_{1}} ... U^*_{j'_{q'}i'_{q'}}dU$$ We give an explicit formula in terms of characters of symmetric groups and Schur functions, which…
In this paper, we derive eight basic identities of symmetry in three variables related to Euler polynomials and alternating power sums. These and most of their corollaries are new, since there have been results only about identities of…
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results…
We apply the asymptotic iteration method (AIM) [J. Phys. A: Math. Gen. 36, 11807 (2003)] to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue…
We present a large number of analytic evaluations of Euler sums, namely sums such as \begin{align} M(m,n_0,n_1,n_2, \ldots, n_t) &= \sum_{k=1}^\infty \frac{H(k)^m}{k^{n_0} (k+1)^{n_1} (k+2)^{n_2} \cdots (k+t)^{n_t}}, \nonumber \end{align}…
It is well known that sometimes Euler sums (i.e., alternating multiple zeta values) can be expressed as $\Q$-linear combinations of multiple zeta values (MZVs). In her thesis Glanois presented a criterion for motivic Euler sums to be…
We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is…
In this paper, we present new explicit simultaneous rational approximations converging sub-exponentially to the values of Bell polynomials at the points of the form $(\gamma, 1! (2a+1)\zeta(2), 2!\zeta(3),..., (m-1)!(a+1+(-1)^ma)\zeta(m)),$…
We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{\nu,p}(a)=\sum_{n\geq 1} (an^p/2)^{-\nu} K_\nu(an^p)\qquad (a>0,\ 0\leq\nu<1)\] as the parameter $a\to 0+$, where $p$…
The purpose of this paper is two-fold. First, we consider the classical Mordell--Tornheim zeta values and their alternating version. It is well-known that these values can be expressed as rational linear combinations of multiple zeta values…