Some generalised extended incomplete beta functions and applications

Yıl 2022, Cilt: 11 Sayı: 2, 132 - 142, 31.08.2022

Oğuz Yağcı , Recep Şahin , İ. Onur Kıymaz , Ayşegül Çetinkaya

https://doi.org/10.54187/jnrs.1143905

Öz

This paper introduces generalised incomplete beta functions defined by the generalised beta function. Firstly, we provide some of the generalised beta function's basic properties, such as integral representations, summation formulas, Mellin transform, and beta distribution. We then present several fundamental properties, such as integral representations, summation formulas, and recurrence relations with the help of the generalised incomplete beta functions.

Anahtar Kelimeler

Gamma function, Beta function, Incomplete beta function, Beta distribution

Kaynakça

• G. E. Andrews, R. Askey, R. Roy, Special Functions. Cambridge University Press, Cambridge, 1999.
• M. A. Chaudhry, S. M. Zubair, On a Class of Incomplete Gamma with Applications. CRC Press (Chapman and Hall), Boca Raton, FL, 2002.
• E. D. Rainville, Special Functions. Macmillan Company, New York, 1960, Reprinted by Chelsea Publishing Company, Bronx, NewYork, 1971.
• R. Şahin, O. Yağcı, Note on the certain special functions representable as $\Phi$, Journal of Fractional Calculus and Applications, 11(2), (2020) 1-11.
• M. A. Chaudhry, A. Qadir, M. Raque, S. M. Zubair, Extension of Euler's beta function, Journal of Computational and Applied Mathematics, 78(1), (1997) 19-32.
• E. Özergin, M. A. Özarslan, A. Altın, Extension of gamma, beta and hypergeometric functions, Journal of Computational and Applied Mathematics, 235(16), (2011) 4601-4610.
• R. K. Parmar, P. Chopra, Generalization of the incomplete extended beta function and beta distribution, International Journal of Engineering Research and Development, 2(4), (2012) 58-62.
• D. Lee, A. K. Rathie, R. K. Parmar, Y. S. Kim, Generalization of extended beta function, hypergeometric and confluent hypergeometric Functions, Honam Mathematical Journal, 33(2), (2011) 187-206.
• J. Choi, A. K. Rathie, R. K. Parmar, Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Mathematical Journal, 33, (2014) 357-385.
• E. Ata, İ. O. Kıymaz, A study on certain properties of generalized special functions defined by Fox-Wright function, Applied Mathematics and Nonlinear Sciences, 5(1), (2020) 147-162.
• A. Fernandez, C. Ustaoğlu, M. A. Özarslan, On the analytical development of incomplete Riemann-Liouville fractional calculus, Turkish Journal of Mathematics, 45(3), (2021) 1418-1443.
• M. J. Luo, G. V. Milovanovic, P. Agarwal, Some results on the extended beta and extended hypergeometric functions, Applied Mathematics and Computation, 248, (2014) 631-651.
• M. A. Özarslan, C. Ustaoğlu, Incomplete Caputo fractional derivative operators, Advances in Difference Equations, 2018, (2018) Article Number: 209, 1-18.
• M. A. Özarslan, C. Ustaoğlu, Extension of incomplete gamma, beta and hypergeometric functions, Progress in Fractional Differentiation and Applications, 5(1), (2019) 21-35.
• M. A. Özarslan, C. Ustaoğlu, Some incomplete hypergeometric functions and incomplete Riemann-Liouville fractional integral operators, Mathematics, 7(5), (2019) 1-17.
• M. A. Özarslan, C. Ustaoğlu, Extended incomplete version of hypergeometric functions, Filomat, 34(2), (2020) 653-662.
• M. A. Özarslan, C. Ustaoğlu, Extended incomplete Riemann-Liouville fractional integral operators and related special functions, Electronic Research Archive, 30(5), (2022), 1723-1747.
• R. K. Parmar, A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions, Le Mathematiche, 68, (2013) 33-52.
• R. Şahin, O. Yağcı, A new generalization of Pochhammer symbol and its applications, Applied Mathematics and Nonlinear Sciences, 5(1), (2020) 255-266.
• R. Şahin, O. Yağcı, Fractional calculus of the extended hypergeometric function, Applied Mathematics and Nonlinear Sciences, 5(1), (2020) 269-284.
• U. M. Abubakar, New generalized beta function associated with the Fox-Wright function, Journal of Fractional Calculus and Application, 12(2), (2021) 204-227.
• U. M. Abubakar, A comparative analysis of modified extended fractional derivative and integral operators via modified extended beta function with applications to generating functions, Çankaya University Journal of Science and Engineering, 19(1), (2022) 40-50.
• U. M. Abubakar, H. M. Tahir, I. S. Abdulmumini, Extended gamma, beta and hypergeometric functions: Properties and applications: Extended gamma, beta and hypergeometric functions, Journal of Kerala Statistical Association, 32(1), (2021) 18-40.
• R. Şahin, O. Yağcı, M. B. Yağbasan, A. Çetinkaya, İ. O. Kıymaz, Further generalizations of gamma, beta and related functions, Journal of Inequalities and Special Functions, 9(4), (2018) 1-7.