scholarly journals Some Inequalities on Generalized p − k Gamma and Beta Functions

2021 ◽  
pp. 61-69
Author(s):  
Inci Ege ◽  
Emrah Yıldırım
Keyword(s):  
Integral Representation ◽  
Beta Function ◽  
Logarithmic Convexity

In this work, we firstly introduce the integral representation of the generalized p-k beta function and then employ well-known Chebychev’s and H¨older’s inequalities to obtain some inequalities on generalized p - k gamma and beta functions. Also, we show the logarithmic convexity propertiesof these functions.

scholarly journals An Extension of Beta Function by Using Wiman’s Function

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 187 ◽  
Author(s):  
Rahul Goyal ◽  
Shaher Momani ◽  
Praveen Agarwal ◽  
Michael Th. Rassias
Keyword(s):  
Integral Representation ◽  
Mellin Transform ◽  
Beta Function ◽  
Functional Relations ◽  
Derivative Formulas ◽  
Extended Beta Function

The main purpose of this paper is to study extension of the extended beta function by Shadab et al. by using 2-parameter Mittag-Leffler function given by Wiman. In particular, we study some functional relations, integral representation, Mellin transform and derivative formulas for this extended beta function.

scholarly journals New Extension of Beta Function and Its Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-25 ◽  
Author(s):  
Mehar Chand ◽  
Hanaa Hachimi ◽  
Rekha Rani
Keyword(s):  
Integral Representation ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Beta Function ◽  
Gauss Hypergeometric Function ◽  
Confluent Hypergeometric Functions ◽  
Integral Formulas ◽  
New Type

In the present paper, new type of extension of classical beta function is introduced and its convergence is proved. Further it is used to introduce the extension of Gauss hypergeometric function and confluent hypergeometric functions. Then we study their properties, integral representation, certain fractional derivatives, and fractional integral formulas and application of these functions.

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.

On an extension of the Hardy-Hilbert theorem

2005 ◽  
Vol 42 (1) ◽  
pp. 21-35 ◽  
Author(s):  
J. Weijian ◽  
G. Mingzhe ◽  
G. Xuemei
Keyword(s):  
Beta Function ◽  
Summation Formula ◽  
Hilbert’S Inequality ◽  
Two Parameters

A weighted Hardy-Hilbert’s inequality with the parameter λ of form \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{m = 1}^\infty {\sum\limits_{n = 1}^\infty {\frac{{a_m b_n }}{{(m + n)^\lambda }}} < B^* (\lambda )\left( {\sum\limits_{n = 1}^\infty {n^{1 - \lambda } a_{a_n }^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \left( {\sum\limits_{n = 1}^\infty {n^{1 - \lambda } b_n^q } } \right)^q }$$ \end{document} is established by introducing two parameters s and λ, where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{1}{p} + \tfrac{1}{q} = 1,p \geqq q > 1,1 - \tfrac{q}{p} < \lambda \leqq 2,B^* (\lambda ) = B(\lambda - (1 - \tfrac{{2 - \lambda }}{p}),1 - \tfrac{{2 - \lambda }}{p})$$ \end{document} is the beta function. B *(λ) is proved to be best possible. A stronger form of this inequality is obtained by means of the Euler-Maclaurin summation formula.

scholarly journals Modeling Diameter Distributions with Six Probability Density Functions in Pinus halepensis Mill. Plantations Using Low-Density Airborne Laser Scanning Data in Aragón (Northeast Spain)

2021 ◽  
Vol 13 (12) ◽  
pp. 2307
Author(s):  
J. Javier Gorgoso-Varela ◽  
Rafael Alonso Ponce ◽  
Francisco Rodríguez-Puerta
Keyword(s):  
Probability Density ◽  
Linear Models ◽  
Pinus Halepensis ◽  
Beta Function ◽  
Probability Density Functions ◽  
Lidar Data ◽  
Density Functions ◽  
Minimum Diameter ◽  
Location Parameters ◽  
Diameter Distributions

The diameter distributions of trees in 50 temporary sample plots (TSPs) established in Pinus halepensis Mill. stands were recovered from LiDAR metrics by using six probability density functions (PDFs): the Weibull (2P and 3P), Johnson’s SB, beta, generalized beta and gamma-2P functions. The parameters were recovered from the first and the second moments of the distributions (mean and variance, respectively) by using parameter recovery models (PRM). Linear models were used to predict both moments from LiDAR data. In recovering the functions, the location parameters of the distributions were predetermined as the minimum diameter inventoried, and scale parameters were established as the maximum diameters predicted from LiDAR metrics. The Kolmogorov–Smirnov (KS) statistic (Dn), number of acceptances by the KS test, the Cramér von Misses (W2) statistic, bias and mean square error (MSE) were used to evaluate the goodness of fits. The fits for the six recovered functions were compared with the fits to all measured data from 58 TSPs (LiDAR metrics could only be extracted from 50 of the plots). In the fitting phase, the location parameters were fixed at a suitable value determined according to the forestry literature (0.75·dmin). The linear models used to recover the two moments of the distributions and the maximum diameters determined from LiDAR data were accurate, with R2 values of 0.750, 0.724 and 0.873 for dg, dmed and dmax. Reasonable results were obtained with all six recovered functions. The goodness-of-fit statistics indicated that the beta function was the most accurate, followed by the generalized beta function. The Weibull-3P function provided the poorest fits and the Weibull-2P and Johnson’s SB also yielded poor fits to the data.

scholarly journals Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 984
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Integral Representation ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Function Spaces ◽  
Laguerre Functions ◽  
Addition Formula ◽  
Lebesgue Spaces ◽  
Rodrigues Formula ◽  
Generalized Laguerre Polynomials ◽  
New Family

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

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