Related papers: On summation formulas in probability theory
Generalized integral formulas involving the generalized Bessel-Maitland function are considered and it expressed in terms of generalized Wright hypergeometric functions. By assuming appropriate values of the parameters in the main results,…
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…
The most general mathematical law for summing bounded quantities is not the arithmetic law, but a composition law of which the summation law for velocities in special relativity is only one particular example. We believe that this…
In the last decades, the theory of digamma function has been developed with a high impact of interest by many authors. Here, we established some interesting results for digamma function, and also we have computed the values of digamma…
We derive special forms of the Poisson summation formula for even and odd functions, which are applied to obtain representations for Euler-type numbers and to sum various series related to elliptic functions.
Using probability theory we derive an expression for the sum of a series of definite integrals involving upper incomplete Gamma functions. In the proof, a normal variance mixture distribution with Beta mixing distributions plays a crucial…
In this work, generalized hypergeometric functions for bicomplex argument is introduced and its convergence criteria is derived. Furthermore, integral representation of this function has been established. Moreover, quadratic transformation,…
A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.
In this paper we present some interesting results involving summation of series in particular trigonometric ones. We failed to locate these results in existing literature or in the web like MathWorld (http://mathworld.wolfram.com/) nor…
In 2021, the first author and Kalita obtained two general hypergeometric formulas for sums involving certain rising factorials to prove some supercongruence conjectures of Guo related to (B.2) and (C.2). In this paper, we further generalize…
Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are…
We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…
We generalize a terminating summation formula to a unilateral nonterminating, and further, a bilateral summation formula by a property of analytic functions. The unilateral one is proved to be a $q$-analogue of a $_4F_3$-summation formula.…
A new class of probability distributions closely connected to generalized hyperbolic distributions is introduced. It is more adapted to study the distributions of sums of random number of random variables. The properties of these…
We present some elementary derivations of summation and transformation formulas for q-series, which are different from, and in several cases simpler or shorter than, those presented in the Gasper and Bahman [1990] "Basic Hypergeometric…
We present recent computer algebra methods that support the calculations of (multivariate) series solutions for (certain coupled systems of partial) linear differential equations. The summand of the series solutions may be built by…
Gaussian Quadrature is a well known technique for numerical integration. Recently Gaussian quadrature with respect to discrete measures corresponding to finite sums have found some new interest. In this paper we apply these ideas to…
In a previous work (arXiv:2505.05574), a summation formula for harmonic Maass forms of polynomial growth was established. In this note, we use the theory of $L$-series of harmonic Maass forms to state and prove a summation formula for such…
In terms of the hypergeometric method, we give the extensions of two known series for $\pi$. Further, other twenty-nine summation formulas for $\pi$, $\pi^2$ and $1/\pi$ with free parameters are also derived in the same way.
Based on some combinatorial identities arising from symbolic summation, we extend two supercongruences on partial sums of hypergeometric series, which were originally conjectured by Guo and Schlosser and recently confirmed by Jana and…