scholarly journals A Note on Superspirals of Confluent Type

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 762 ◽  
Author(s):  
Jun-ichi Inoguchi ◽  
Rushan Ziatdinov ◽  
Kenjiro T. Miura
Keyword(s):  
Hypergeometric Function ◽  
Confluent Hypergeometric Function ◽  
Gauss Hypergeometric Function ◽  
Radius Of Curvature ◽  
Planar Curves ◽  
Completely Monotonic ◽  
New Class ◽  
Monotone Curvature ◽  
Similarity Geometry

Superspirals include a very broad family of monotonic curvature curves, whose radius of curvature is defined by a completely monotonic Gauss hypergeometric function. They are generalizations of log-aesthetic curves, and other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. In this work, we study superspirals of confluent type via similarity geometry. Through a detailed investigation of the similarity curvatures of superspirals of confluent type, we find a new class of planar curves with monotone curvature in terms of Tricomi confluent hypergeometric function. Moreover, the proposed ideas will be our guide to expanding superspirals.

scholarly journals Analogues of the general theta transformation formula

2013 ◽  
Vol 143 (2) ◽  
pp. 371-399 ◽  
Author(s):  
Atul Dixit
Keyword(s):  
Hypergeometric Function ◽  
Infinite Series ◽  
Confluent Hypergeometric Function ◽  
Möbius Function ◽  
Transformation Formula ◽  
Extended Version ◽  
New Class ◽  
Mobius Function ◽  
By Products

A new class of integrals involving the confluent hypergeometric function 1F1(a;c;z) and the Riemann Ξ-function is considered. It generalizes a class containing some integrals of Ramanujan, Hardy and Ferrar and gives, as by-products, transformation formulae of the form F(z, α) = F(iz, β), where αβ = 1. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulae of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. We conclude with a generalization of a conjecture due to Ramanujan, Hardy and Littlewood involving infinite series of the Möbius function.

scholarly journals Some Inequalities of Extended Hypergeometric Functions

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2702
Author(s):  
Shilpi Jain ◽  
Rahul Goyal ◽  
Praveen Agarwal ◽  
Juan L. G. Guirao
Keyword(s):  
Hypergeometric Function ◽  
Integral Inequality ◽  
Hypergeometric Functions ◽  
Beta Function ◽  
Confluent Hypergeometric Function ◽  
Gauss Hypergeometric Function ◽  
Extended Type ◽  
Mathematical Sciences ◽  
Set Up ◽  
Extended Beta Function

Hypergeometric functions and their inequalities have found frequent applications in various fields of mathematical sciences. Motivated by the above, we set up certain inequalities including extended type Gauss hypergeometric function and confluent hypergeometric function, respectively, by virtue of Hölder integral inequality and Chebyshev’s integral inequality. We also studied the monotonicity, log-concavity, and log-convexity of extended hypergeometric functions, which are derived by using the inequalities on an extended beta function.

scholarly journals A study on certain properties of generalized special functions defined by Fox-Wright function

2020 ◽  
Vol 5 (1) ◽  
pp. 147-162
Author(s):  
Enes Ata ◽  
İ. Onur Kıymaz
Keyword(s):  
Hypergeometric Function ◽  
Special Functions ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Summation Formulas ◽  
Frequent Use ◽  
Derivative Formulas ◽  
Transformation Formulas

AbstractIn this study, motivated by the frequent use of Fox-Wright function in the theory of special functions, we first introduced new generalizations of gamma and beta functions with the help of Fox-Wright function. Then by using these functions, we defined generalized Gauss hypergeometric function and generalized confluent hypergeometric function. For all the generalized functions we have defined, we obtained their integral representations, summation formulas, transformation formulas, derivative formulas and difference formulas. Also, we calculated the Mellin transformations of these functions.

HYPERGEOMETRIC CAUCHY NUMBERS

2013 ◽  
Vol 09 (03) ◽  
pp. 545-560 ◽  
Author(s):  
TAKAO KOMATSU
Keyword(s):  
Positive Integer ◽  
Hypergeometric Function ◽  
Confluent Hypergeometric Function ◽  
Bernoulli Numbers ◽  
Gauss Hypergeometric Function ◽  
Cauchy Numbers ◽  
Sums Of Products

For a positive integer N, define hypergeometric Cauchy numbers cN,n by [Formula: see text] where 2 F1(a, b;c;z) is the Gauss hypergeometric function. When N = 1, c1,n = cn are classical Cauchy numbers. In this paper we shall consider sums of products of hypergeometric Cauchy numbers. We note that hypergeometric Cauchy numbers are analogous to hypergeometric Bernoulli numbers BN,n defined by [Formula: see text] where 1F1(a;b;z) is the confluent hypergeometric function.

scholarly journals Some Results of Extended Beta Function and Hypergeometric Functions by Using Wiman’s Function

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2944
Author(s):  
Shilpi Jain ◽  
Rahul Goyal ◽  
Praveen Agarwal ◽  
Antonella Lupica ◽  
Clemente Cesarano
Keyword(s):  
Hypergeometric Function ◽  
Beta Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Gauss Hypergeometric Function ◽  
Logarithmic Convexity ◽  
Functional Relations ◽  
Derivative Formulas ◽  
Extended Beta Function ◽  
Transformation Formulas

The main aim of this research paper is to introduce a new extension of the Gauss hypergeometric function and confluent hypergeometric function by using an extended beta function. Some functional relations, summation relations, integral representations, linear transformation formulas, and derivative formulas for these extended functions are derived. We also introduce the logarithmic convexity and some important inequalities for extended beta function.

scholarly journals The Bateman Functions Revisited after 90 Years—A Survey of Old and New Results

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1273
Author(s):  
Alexander Apelblat ◽  
Armando Consiglio ◽  
Francesco Mainardi
Keyword(s):  
Hypergeometric Function ◽  
Laplace Transforms ◽  
Confluent Hypergeometric Function ◽  
Integral Function ◽  
Basic Properties ◽  
Differential Relations

The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions, their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to generalizations of these functions and to the Bateman-integral function known in the literature.

Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity, I

1984 ◽  
Vol 99 (1-2) ◽  
pp. 51-70 ◽  
Author(s):  
F. V. Atkinson ◽  
C. T. Fulton
Keyword(s):  
Asymptotic Expansion ◽  
Eigenvalue Problem ◽  
Hypergeometric Function ◽  
Finite Interval ◽  
Confluent Hypergeometric Function ◽  
Limit Circle ◽  
Asymptotic Formulae ◽  
Sturm Liouville

SynopsisAsymptotic formulae for the positive eigenvalues of a limit-circle eigenvalue problem for –y” + qy = λy on the finite interval (0, b] are obtained for potentials q which are limit circle and non-oscillatory at x = 0, under the assumption xq(x)∈L1(0,6). Potentials of the form q(x) = C/xk, 0<fc<2, are included. In the case where k = 1, an independent check based on the limit-circle theory of Fulton and an asymptotic expansion of the confluent hypergeometric function, M(a, b; z), verifies the main result.

INTEGRAL TRANSFORM OF K4-FUNCTION

2021 ◽  
Vol 21 (2) ◽  
pp. 429-436
Author(s):  
SEEMA KABRA ◽  
HARISH NAGAR
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Integral Transform ◽  
Integral Transforms ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Generalized Wright Function ◽  
Special Cases ◽  
Euler Transform

In this present work we derived integral transforms such as Euler transform, Laplace transform, and Whittaker transform of K4-function. The results are given in generalized Wright function. Some special cases of the main result are also presented here with new and interesting results. We further extended integral transforms derived here in terms of Gauss Hypergeometric function.

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