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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 is an m th power residue modulo p } . In particular, if p = x 2 + 64 y 2 with x , y , then k R 4 ( p ) 1 + tan π a k p = ( - 1 ) y ( - 2 ) ( p - 1 ) / 8 .

Theorem for Series in Three-Parameter Mittag-Leffler Function

Soubhia, Ana, Camargo, Rubens, Oliveira, Edmundo, Vaz, Jayme (2010)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification 2010: 26A33, 33E12.The new result presented here is a theorem involving series in the three-parameter Mittag-Leffler function. As a by-product, we recover some known results and discuss corollaries. As an application, we obtain the solution of a fractional differential equation associated with a RLC electrical circuit in a closed form, in terms of the two-parameter Mittag-Leffler function.

Theorems on the Convergence of Series in Generalized Lommel-Wright Functions

Paneva-Konovska, Jordanka (2007)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 30B10, 30B30; 33C10, 33C20The classical Cauchy-Hadamard, Abel and Tauber theorems provide useful information on the convergence of the power series in complex plane. In this paper we prove analogous theorems for series in the generalized Lommel-Wright functions with 4 indices. Results for interesting special cases of series involving Bessel, Bessel-Maitland, Lommel and Struve functions, are derived.We provide also a new asymptotic formula for the generalized ...

Theoretical and numerical studies of the P N P M DG schemes in one space dimension

Abdulatif Badenjki, Gerald G. Warnecke (2019)

Applications of Mathematics

We give a proof of the existence of a solution of reconstruction operators used in the P N P M DG schemes in one space dimension. Some properties and error estimates of the projection and reconstruction operators are presented. Then, by applying the P N P M DG schemes to the linear advection equation, we study their stability obtaining maximal limits of the Courant numbers for several P N P M DG schemes mostly experimentally. A numerical study explains how the stencils used in the reconstruction affect the efficiency...

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