相关论文: Summed series involving \,_{1}F_{2} hypergeometric…
A summation formula is derived for the sum of the first m+1 terms of the 3F2(a,b,c;(a+b+1)/2,2c;1) series when c = -m is a negative integer. This summation formula is used to derive a formula for the sum of a terminating double…
Several sums of Neumann series with Bessel and trigonometric functions are evaluated, as finite sums of trigonometric functions. They arise from a generalization of the Neumann expansion of the eigenstates of the Laplacian in regular…
This paper considers functional series whose terms are higher-order derivatives of Chebyshev polynomials of the second kind, where the degree of the polynomial is related to the order of the derivative. Analytic summation is used to…
We examine convergent representations for the sum of Bessel functions \[\sum_{n=1}^\infty \frac{J_\mu(na) J_\nu(nb)}{n^{\alpha}}\] for $\mu$, $\nu\geq0$ and positive values of $a$ and $b$. Such representations enable easy computation of the…
We derive a convergent expansion of the generalized hypergeometric function ${}_{p-1}F_p$ in terms of the Bessel functions ${}_{0}F_1$ that holds uniformly with respect to the argument in any horizontal strip of the complex plane. We…
Results are presented for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. The sums of these series can be evaluated…
The special case of the hypergeometric function $_{2}F_{1}$ represents the binomial series $(1+x)^{\alpha}=\sum_{n=0}^{\infty}(\:\alpha n\:)x^{n}$ that always converges when $|x|<1$. Convergence of the series at the endpoints, $x=\pm 1$,…
An extension of two finite trigonometric series is studied to derive closed form formulae involving the Hurwitz-Lerch zeta function. The trigonometric series involves angles with a geometric series involving the powers of 3. These closed…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
Let $F(n,k)$ be a hypergeometric function that may be expressed so that $n$ appears within initial arguments of inverted Pochhammer symbols, as in factors of the form $\frac{1}{(n)_{k}}$. Only in exceptional cases is $F(n, k)$ such that…
We prove a summation formula for a bilateral series whose terms are products of two basic hypergeometric functions. In special cases, series of this type arise as matrix elements of quantum group representations.
We express the asymptotics of the remainders of the partial sums {s_n} of the generalized hypergeometric function q+1_F_q through an inverse power series z^n n^l \sum_k c_k/n^k, where the exponent l and the asymptotic coefficients {c_k} may…
A new formula is derived that generalises an earlier result for the infinite integral over three spherical Bessel functions. The analytical result involves a finite sum over associated Legendre functions, $P_l^m(x)$, of degree $l$ and order…
Analytic expressions for the Fourier transforms of the Chebyshev and Legendre polynomials are derived, and the latter is used to find a new representation for the half-order Bessel functions. The numerical implementation of the so-called…
Recent progress in analytical calculation of the multiple [inverse, binomial, harmonic] sums, related with epsilon-expansion of the hypergeometric function of one variable are discussed.
In this paper, sums represented in (3) are studied. The expressions are derived in terms of Bessel functions of the first and second kinds and their integrals. Further, we point out the integrals can be written as a Meijer G function.
The generalized hypergeometric function $_qF_p$ is a power series in which the ratio of successive terms is a rational function of the summation index. The Gaussian hypergeometric functions $_2F_1$ and $_3F_2$ are most common special cases…
Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers…
We express some general type of infinite series such as $$ \sum^\infty_{n=1}\frac{F(H_n^{(m)}(z),H_n^{(2m)}(z),\ldots,H_n^{(\ell m)}(z))} {(n+z)^{s_1}(n+1+z)^{s_2}\cdots (n+k-1+z)^{s_k}}, $$ where $F(x_1,\ldots,x_\ell)\in\mathbb…
The present note considers a certain family of sums indexed by the set of fixed length compositions of a given number. The sums in question cannot be realized as weighted compositions. However they can be be related to the hypergeometric…