Abstract
The main focus of this paper is to investigate some new properties of the incomplete Fox–Wright function. We derive an integral representation of the incomplete Fox–Wright function whose terms contain Fox’s H-function. As a direct consequence, it leads to some new results including inequalities, log-convexity and complete monotonicity for this function. Moreover, it yields interesting monotonicity involving the ratios of the incomplete Fox–Wright function. Furthermore, certain generating functions for the incomplete Fox–Wright function when their terms contain the Appell function of the first kind are established. Finally, by means of the generating functions obtained here, new summation formula for the incomplete Fox–Wright function in terms of the H-function of two variables is made.
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References
Askey, R.A., Olde Daalhuis, A.B.: Generalized hypergeometric functions and Meijer \(G\)-function. In: Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.) NIST Handbook of Mathematical Functions. Cambridge University Press, NewYork (2010)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. I. McGraw-Hill Book Company, New York (1953)
Fox, Ch.: The asymptotic expansion of generalized hypergeometric functions. Proc. Lond. Math. Soc. S2–27(1), 389–400 (1928)
Karp, D., Prilepkina, E.: Completely monotonic gamma ratio and infinitely divisible \(H\)-function of Fox. Comput. Methods Funct. Theory 16(1), 135–153 (2016)
Kimberling, C.H.: A probabilistic interpretation of complete monotonicity. Aequationes Math. 10, 152–164 (1974)
Mathai, A.M., Saxena, R.K., Haubold, H.J.: The \(H\)-Functions: Theory and Applications. Springer, New York (2010)
Mehrez, K.: New integral representations for the Fox–Wright functions and its applications. J. Math. Anal. Appl. 468, 650–673 (2018)
Mehrez, K.: Some geometric properties of a class of functions related to the Fox–Wright functions. Banach J. Math. Anal. 14, 1222–1240 (2020)
Mehrez, K.: New integral representations for the Fox-Wright functions and its applications II. J. Contemp. Math. Anal. 56(1), 37–45 (2021)
Mehrez, K., Sitnik, S.M.: Functional inequalities for the Fox-Wright functions. Ramanujan J. 50(2), 263–287 (2019)
Mitrinović, D.S.: Analytic Inequalities. Springer-Verlag, Berlin (1970)
Mittal, P.K., Gupta, K.C.: An integral involving generalized function of two variables. Proc. Indian Acad. Sci. A 75, 117–123 (1972)
Neuman, E.: Inequalities and bounds for the incomplete Gamma function. Results Math. 63, 1209–1213 (2013)
Prabhakar, T.R.: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)
Qi, F.: Monotonicity results and inequalities for the gamma and incomplete gamma functions. Math. Inequal. Appl. 5(1), 61–67 (2002)
Saigo, M., Saxena, R.K.: Applications of generalized fractional calculus operators in the solution of an integral equation. J Frac Calc. 14, 53–63 (1998)
Saxena, R.K., Nishimoto, K.: Fractional integral formula for the H-function. J. Fract. Calc. 6, 65–75 (1994)
Srivastava, H.M., Saxena, R.K., Parmar, R.K.: Some families of the incomplete \(H\)-functions and the incomplete \({\bar{H}}\)-functions and associated integral transforms and operators of fractional calculus with applications. Russ. J. Math. Phys. 25(1), 116–138 (2018)
Srivastava, H.M., Gupta, K.C., Goyal, S.P.: The \(H\)-Functions of One and Two Variables with Applications. South Asian Publishers, New Delhi (1982)
Wright, E.M.: The asymptotic expansion of the generalized hypergeometric function. J. Lond. Math. Soc. 10, 287–293 (1935)
Wright, E.M.: The asymptotic expansion of the generalized hypergeometric function. Proc. Lond. Math. Soc. (Ser. 2) 46, 389–408 (1940)
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Dedicated to my children Youssef and Zina.
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Mehrez, K. Integral representation and computational properties of the incomplete Fox–Wright function. Ramanujan J 58, 369–387 (2022). https://doi.org/10.1007/s11139-022-00571-7
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DOI: https://doi.org/10.1007/s11139-022-00571-7