Ramanujan's cubic transformation inequalities for zero-balanced hypergeometric functions

On applications of hypergeometric functions

Sharp Landen transformation inequalities for hypergeometric functions, with applications

An approximation for zero-balanced Appell function [formula omitted] near [formula omitted

The authors study certain monotoneity and convexity properties of the Gaussian hypergeometric function and those of the Euler gamma function.

Log-convexity and log-concavity of hypergeometric-like functions

Turán type inequalities for [formula omitted]-hypergeometric functions

Harmonic Analysis and Special Functions on Symmetric Spaces

Based on the Hadamard product of power series, polynomial series expansions for confluent hypergeometric functions M ( a , c ; ⋅ ) M(a,c;\cdot ) and for Gaussian hypergeometric functions F ( a , b ; c ; ⋅ ) F(a,b;c;\cdot ) are introduced and studied. It turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions M ( a , c ; ⋅ ) M(a,c;\cdot ) and F ( a , b ; c ; ⋅ ) F(a,b;c;\cdot ) , in particular, if the parameters are also viewed as variables.

Use of the Gaussian hypergeometric function to solve the equation of gradually-varied flow

In this paper we first discuss refinement of the Ramunujan asymptotic expansion for the classical hypergeometric functionsF(a,b;c;x), c ≤a + b, near the singularityx = 1. Further, we obtain monotonous properties of the quotient of two hypergeometric functions and inequalities for certain combinations of them. Finally, we also solve an open problem of finding conditions ona, b > 0 such that 2F(−a,b;a +b;r 2) < (2−r 2)F(a,b;a +b;r 2) holds for all r∈(0,1).

This paper demonstrates several of Ostrowski-type inequalities for fuzzy number functions and investigates their connections with other inequalities. Specifically, employing the Aumann integral and the Kulisch–Miranker order, as well as the inclusion order on the space of real and compact intervals, we establish various Ostrowski-type inequalities for fuzzy-valued mappings (F·V·Ms). Furthermore, by employing diverse orders, we establish connections with the classical versions of Ostrowski-type inequalities. Additionally, we explore new ideas and results rooted in submodular measures, accompanied by examples and applications to illustrate our findings. Moreover, by using special functions, we have provided some applications of Ostrowski-type inequalities.

Modelling using an extended Yule distribution

We provide formulas for traces of p t h p^{th} Hecke operators in level 1 in terms of values of finite field 2 F 1 _2F_1 -hypergeometric functions, extending previous work of the author to all odd primes p p instead of only those p ≡ 1 ( mod 12 ) p \equiv 1 \pmod {12} . We first give a general level 1 trace formula in terms of the trace of Frobenius on a family of elliptic curves, and then we draw on recent work of Lennon to produce level 1 trace formulas in terms of hypergeometric functions for all primes p &gt; 3 p &gt; 3 .

In this paper, we investigate the channel capacity of an additive Bernoulli-Gaussian (BG) impulsive noise channel in Rayleigh fading via lower and upper bounds. To this end, we first show that the differential entropy of the BG impulse noise can be established in closed-form using Gaussian hypergeometric function <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (1, 1; .; .). This closed-form expression allows us to derive a lower bound on the capacity limit obtained by a Gaussian input using the Gauss-Hermite quadrature formula. We also derive in closed-form two upper bounds on the channel capacity. The first upper bound is obtained under the assumption of full knowledge of noise state, while the second upper bound is developed using a Gaussian distributed output. At high power regions, the lower bound achieved by Gaussian inputs and the upper bound generated by Gaussian inputs are indistinguishable. These two bounds can therefore be used as an accurate estimation for the channel capacity. When the channel input power is small compared to the power of the impulsive noise component, the lower bound obtained by using a Gaussian input and the upper bound under the perfect knowledge of impulse noise state are almost identical, which are useful to predict the capacity. The establishment of the lower bound and the two upper bounds in closed-form helps us to confirm the near-optimality of the Gaussian input in a wide range of input power levels over BG impulsive noise channels in Rayleigh fading.

We derive an expression for the value ζQ(3) of the spectral zeta function ζQ(s) for the non-commutative harmonic oscillator using a Gaussian hypergeometric function. In this study, two sequences of rational numbers, denoted ${ ilde J}\\sb 2 (n)$ and ${ ilde J}\\sb 3 (n)$, which can be regarded as analogues of the Apery numbers, naturally arise and play a key role in obtaining the expressions for the values ζQ(2) and ζQ(3). We also show that the numbers ${ ilde J}\\sb 2 (n)$ and ${ ilde J}\\sb 3 (n)$ have congruence relations such as those of the Apery numbers.