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

The transcendence of certain quasi-periods associated with abelian functions in two variables

D. W. Masser (1977)

Compositio Mathematica

The triplication formula for Gauss sums.

John Greene, Dennis Stanton (1986)

Aequationes mathematicae

The truncated exponential polynomials, the associated Hermite forms and applications.

Dattoli, G., Migliorati, M. (2006)

International Journal of Mathematics and Mathematical Sciences

The Weyl fractional operator of a system of polynomials

R. K. Raina (1986)

Rendiconti del Seminario Matematico della Università di Padova

The zeros of $a z^{2} J_{ν}^{''} (z) + b z J_{ν}^{'} (z) + c J_{ν} (z)$ as functions of order.

Mercer, A.McD. (1992)

International Journal of Mathematics and Mathematical Sciences

The zeros of exponential polynomials (I)

Carlos Julio Moreno (1973)

Compositio Mathematica

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 for double-index one-parameter Bessel functions.

Castillo, Greilyn, Galué, Leda (2010)

Revista Colombiana de Matemáticas

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...

Théorie analytique du logarithme népérien et de la fonction exponentielle

Ch. Méray (1890)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Theorie der elliptischen Functionen [Book]

Heinrich Durège (1878)

Théorie des fonctions elliptiques [Book]

C. A. A. Briot, J. C. Bouquet (1875)

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Displaying 81 – 100 of 140

The summation package Sigma: underlying principles and a rhombus tiling application.

The symmetrical $H_{q}$ -semiclassical orthogonal polynomials of class one.

The $(t, q)$ -analogs of secant and tangent numbers.

The tangent function and power residues modulo primes

The transcendence of certain quasi-periods associated with abelian functions in two variables

The triplication formula for Gauss sums.

The truncated exponential polynomials, the associated Hermite forms and applications.

The Weyl fractional operator of a system of polynomials

The zeros of $a z^{2} J_{ν}^{''} (z) + b z J_{ν}^{'} (z) + c J_{ν} (z)$ as functions of order.

The zeros of exponential polynomials (I)

Theorem for Series in Three-Parameter Mittag-Leffler Function

Theorems for double-index one-parameter Bessel functions.

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

Theoretical and numerical studies of the $P_{N} P_{M}$ DG schemes in one space dimension

Théorie analytique du logarithme népérien et de la fonction exponentielle

Theorie der elliptischen Functionen [Book]

Théorie des fonctions elliptiques [Book]

Théorie élémentaire des fonctions elliptiques

Théorie élémentaire des fonctions elliptiques

Théorie élémentaire des fonctions elliptiques

Currently displaying 81 – 100 of 140