Two closed-form evaluations for the generalized hypergeometric function 4F3(116){}_4F_3(\frac1{16})

The objective of this short note is to provide two closed-form evaluations for the generalized hypergeometric function $_4F_3$ of the argument $\frac1{16}$. This is achieved by means of separating a generalized hypergeometric function $_3F_2$ into even and odd components, together with the use of two known results for $_3F_2(\pm\frac14)$ available in the literature. As an application, we obtain an interesting infinite-sum representation for the number $\pi^2$. Certain connections with the work of Ramanujan and other authors are discussed, involving other special functions and binomial sums of different kinds

A Feynman integral in Lifshitz-point and Lorentz-violating theories in R^D ⨁ R<i>^m</i>

We evaluate a 1-loop, 2-point, massless Feynman integral ID,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum RD ⨁ Rm . Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of ID,m(p,q) in terms of powers of the variable X:=4p2/q4, where p=|p|, q=|q|, p Є RD, q Є Rm, and in terms of generalised hypergeometric functions 3F2(−X), when X&lt;1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations

Geometric Properties of Generalized Hypergeometric Functions

In this article, Using Hadamard product for $_4F_3\left(^{a_1,\, a_2,\, a_3,\, a_4}_{b_1,\, b_2,\, b_3};z\right)$ hypergeometric function with normalized analytic functions in the open unit disc, an operator $\mathcal{I}^{a_1,a_2,a_3,a_4}_{b_1,b_2,b_3}(f)(z)$ is introduced. Geometric properties of $_4F_3\left(^{a_1,\, a_2,\, a_3,\, a_4}_{b_1,\, b_2,\, b_3};z\right)$ hypergeometric functions are discussed for various subclasses of univalent functions. Also, we consider an operator $\mathcal{I}^{ a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4} }_{ \frac{c}{4}, \frac{c+1}{4}, \frac{c+2}{4},\frac{c+3}{4} }(f)(z)$= z\, _5F_4\left(^{a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4}}_{\frac{c}{4}, \frac{c+1}{4}, \frac{c+2}{4},\frac{c+3}{4}}; z\right)*f(z), where, $_5F_4(z)$ hypergeometric function and the $*$ is usual Hadamard product. In the main results, conditions are determined on $a,b,$ and $c$ such that the function z\, _5F_4\left(^{a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4}}_{\frac{c}{4}, \frac{c+1}{4}, \frac{c+2}{4},\frac{c+3}{4}}; z\right) is in the each of the classes $\mathcal{S}^{*}_{\lambda}$, $\mathcal{C}_{\lambda}$, $UCV$ and $\mathcal{S}_p$. Subsequently, conditions on $a,\,b,\,c,\, \lambda,$ and $\beta$ are determined using the integral operator such that functions belonging to $\mathcal{R}(\beta)$ and $\mathcal{S}$ are mapped onto each of the classes $\mathcal{S}^*_\lambda$, $\mathcal{C}_{\lambda}$, $UCV$, and $\mathcal{S}_p$.Comment: 44 Pages. arXiv admin note: text overlap with arXiv:2205.1338

Some Additions to a Family of Sums and Integrals related to Hurwitz' Zeta Function(s), Euler polynomials and Euler Numbers

Integrals involving the kernel function $sech (\pi x)$ over a semi-infinite range are of general interest in the study of Riemann's function $\zeta(s)$ and Hurwitz' function $\zeta(s,a)$. Such integrals that include the $arctan$ and $log$ functions in the integrand are evaluated here in terms of $\zeta(s,a)$, thereby adding some new members to a known family of related integrals. A claimed connection between $\zeta(s)$ of odd integer argument and such integrals is verified.Comment: Changed title; improved notation; added new section 3.1.