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The paper addresses the study and applications of a broad class of extended-real-valued functions, known as optimal value or marginal functions, which are frequently appeared in variational analysis, parametric optimization, and a variety…
In this paper, additional properties of the lower gamma functions and the error functions are introduced and proven. In particular, we prove interesting relations between the error functions and Laplace transform.
In this paper, we prove some inequalities for the differences and ratios of the beta function.
The suggested approach is based on a known representation of Dirichlet $L$-functions via the incomplete gamma functions. Some properties of the Taylor coefficients of the lower incomplete gamma function at infinity seem to be new.…
In this article, we introduce incomplete Lauricella matrix functions (ILMFs) of $n$ variables through application of the incomplete Pochhammer matrix symbols. Furthermore there is derivation of certain properties; matrix differential…
Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $L$-functions to a power-full modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable square-root of the…
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…
In this paper we first establish new explicit estimates for Chebyshev's $\vartheta$-function. Applying these new estimates, we derive new upper and lower bounds for some functions defined over the prime numbers, for instance the prime…
An analysis of the zeta and gamma function is presented, using elementary functions like [] and {}, a general formula for the angle of zeta(1/2 + i*n) is found and the same for the gamma function.
We examine the use of the Euler-Maclaurin formula and new derived uniform asymptotic expansions for the numerical evaluation of the Lerch transcendent $\Phi(z, s, a)$ for $z, s, a \in \mathbb{C}$ to arbitrary precision. A detailed analysis…
Contour integral representations for Riemann's Zeta function and Dirichelet's Eta (alternating Zeta) function are presented and investigated. These representations flow naturally from methods developed in the 1800's, but somehow they do not…
Some integrals of the Glaisher-Ramanujan type are established in a more general form than in previous studies. As an application we prove some Ramanujan-type series identities, as well as a new formula for the Dirichlet beta function at the…
We introduce new generalizations of the Gamma and the Beta functions. Their properties are investigated and known results are obtained as particular cases.
A simple and elementary derivation of values at integer points for the Riemann's zeta and related functions is reported.
We prove general fomulas for the deviations of two overpartition ranks from the average. These formulas are in terms of Appell--Lerch series and sums of quotients of theta functions and can be used, among other things, to recover any of the…
We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank…
We prove new relations on zeta function at even arguments and Dirichlet $L$ function at odd. The key idea is to make use of the Taylor series and partial fraction decomposition of cotangent and secant functions as we discuss in calculus and…
Using a different approach, we derive integral representations for the Riemann zeta function and its generalizations (the Hurwitz zeta, $\zeta(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent,…
We describe in detail three distinct families of generalized zeta functions built over the (nontrivial) zeros of a rather general arithmetic zeta or L-function, extending the scope of two earlier works that treated the Riemann zeros only.…
Inspired by certain interesting recent extensions of the gamma, beta and hypergeometric matrix functions, we introduce here new extension of the gamma and beta matrix function. We also introduce new extensions of the Gauss hypergeometric…