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The Deformed Trigonometric Functions of two Variables

Marinkovic, Sladjana, Stankovic, Miomir, Mulalic, Edin (2012)

Mathematica Balkanica New Series

MSC 2010: 33B10, 33E20Recently, various generalizations and deformations of the elementary functions were introduced. Since a lot of natural phenomena have both discrete and continual aspects, deformations which are able to express both of them are of particular interest. In this paper, we consider the trigonometry induced by one parameter deformation of the exponential function of two variables eh(x; y) = (1 + hx)y=h (h 2 R n f0g, x 2 C n f¡1=hg, y 2 R). In this manner, we define deformed sine...

The multiple gamma function and its q-analogue

Kimio Ueno, Michitomo Nishizawa (1997)

Banach Center Publications

We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vignéras multiple gamma function by considering the classical limit of the multiple q-gamma function.

The permutation group method for the dilogarithm

Georges Rhin, Carlo Viola (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We give qualitative and quantitative improvements on all the best previously known irrationality results for dilogarithms of positive rational numbers. We obtain such improvements by applying our permutation group method to the diophantine study of double integrals of rational functions related to the dilogarithm.

The reciprocal of the beta function and G L ( n , ℝ ) Whittaker functions

Eric Stade (1994)

Annales de l'institut Fourier

In this paper we derive, using the Gauss summation theorem for hypergeometric series, a simple integral expression for the reciprocal of Euler’s beta function. This expression is similar in form to several well-known integrals for the beta function itself.We then apply our new formula to the study of G L ( n , ℝ ) Whittaker functions, which are special functions that arise in the Fourier theory for automorphic forms on the general linear group. Specifically, we deduce explicit integral representations of “fundamental”...

The tangent function and power residues modulo primes

Zhi-Wei Sun (2023)

Czechoslovak Mathematical Journal

Let p be an odd prime, and let a be an integer not divisible by p . When m is a positive integer with p ≡ 1 ( mod 2 m ) and 2 is an m th power residue modulo p , we determine the value of the product ∏ k ∈ R m ( p ) ( 1 + tan ( π a k / p ) ) , where R m ( p ) = { 0 < k < p : k ∈ ℤ is an m th power residue modulo p } . In particular, if p = x 2 + 64 y 2 with x , y ∈ ℤ , then ∏ k ∈ R 4 ( p ) 1 + tan π a k p = ( - 1 ) y ( - 2 ) ( p - 1 ) / 8 .

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