A Lambda Analogue of the Gamma Function and its Properties

K. Nantomah; I. Ege

doi:10.15421/242209

A Lambda Analogue of the Gamma Function and its Properties

K. Nantomah (C.K. Tedam University of Technology and Applied Sciences)
I. Ege (Aydin Adnan Menderes University)

Abstract

We consider a generalization of the gamma function which we term as lambda analogue of the gamma function or $$$\lambda$$$-gamma function and further, we establish some of its accompanying properties. For the particular case when $$$\lambda=1$$$, the results established reduce to results involving the classical gamma function. The techniques employed in proving our results are analytical in nature.

Keywords

Gamma function; lambda analogue; $$$\lambda$$$-gamma function; $$$\lambda$$$-beta function; Bohr-Mollerup theorem; inequality

MSC 2020

33B15; 26A48; 26D07; 33B20

Full Text:

PDF

References

Abramowitz M., Stegun I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1965.

Al-Jararha M.M., Al-Jararha J.M. "Inequalities of Gamma Function Appearing in Generalizing Probability Sampling Designs", Aust. J. Math. Anal. Appl., 2020; 17(2): Art. 17, p. 9.

Chaudhry M.A., Zubair S.M. "Generalized incomplete gamma functions with applications", J. Comput. Appl. Math., 1994; 55: pp. 99-124. doi:10.1016/0377-0427(94)90187-2

Diaz R., Pariguan E. "On hypergeometric functions and Pochhammer $$$k$$$-symbol", Divulg. Mat., 2007; 15: pp. 179-192.

Diaz R., Teruel C. "$$$q,k$$$-generalized gamma and beta functions", J. Nonlinear Math. Phys., 2005; 12(1): pp. 118-134. doi:10.2991/jnmp.2005.12.1.10

Djabang E., Nantomah K., Iddrisu M.M. "On a v-analogue of the Gamma function and some associated inequalities", J. Math. Comput. Sci., 2020; 11(1): pp. 74-86.

Fu H., Peng Y., Du T. "Some inequalities for multiplicative tempered fractional integrals involving the $$$\lambda$$$-incomplete gamma functions", AIMS Mathematics, 2021; 6(7): pp. 7456-7478. doi:10.3934/math.2021436

Gautschi W. "A harmonic mean inequality for the gamma function", SIAM J. Math. Anal., 1974; 5: pp. 278-281. doi:10.1137/0505030

Grenie L., Molteni G. "Inequalities for the Beta function", Math. Inequal. Appl., 2015; 18(4): pp. 1427-1442. doi:10.7153/mia-18-111

Krasniqi V., Merovci F. "Some Completely Monotonic Properties for the (p, q)-Gamma Function", Math. Balkanica (N.S.), 2012; 26: pp. 1-2.

Loc T.G., Tai T. "The Generalized Gamma Functions", Acta Math. Vietnam., 2012; 37(2): pp. 219-230.

Mohammed P.O., Sarikaya M.Z., Baleanu D. "On the Generalized Hermite-Hadamard Inequalities via the Tempered Fractional Integrals", Symmetry, 2020; 12(4), 595: pp. 1-17. doi:10.3390/sym12040595

Nantomah K., Prempeh E., Twum S.B. "On a $$$(p,k)$$$-analogue of the Gamma function and some associated Inequalities", Moroccan J. of Pure and Appl. Anal., 2016; 2(2): pp. 79-90. doi:10.7603/s40956-016-0006-0

Qi F. "Bounds for the Ratio of Two Gamma Functions", J. Inequal. Appl., 2010; 493058. doi:10.1155/2010/493058

Sandor J. "On certain inequalities for the Gamma function", RGMIA Res. Rep. Coll., 2006; 9(1), Art. 11.