Abstract
In the present paper we derive three interesting expressions for the composition of two most general fractional integral oprators whose kernels involve the product of a general class of polynomials and a multivariableH-function. By suitably specializing the coefficients and the parameters in these functions we can get a large number of (new and known) interesting expressions for the composition of fractional integral operators involving classical orthogonal polynomials and simpler special functions (involving one or more variables) which occur rather frequently in problems of mathematical physics. We have mentioned here two special cases of the first composition formula. The first involves product of a general class of polynomials and the Fox’sH-functions and is of interest in itself. The findings of Buschman [1] and Erdélyi [4] follow as simple special cases of this composition formula. The second special case involves product of the Jacobi polynomials, the Hermite polynomials and the product of two multivariableH-functions. The present study unifies and extends a large number of results lying scattered in the lierature. Its findings are general and deep.
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References
Buschman R G, Fractional integration,Math. Jpn. 9 (1964) 99–106
Erdélyi A, On fractional integration and its application to the theory of Hankel transforms,Quart. J. Math. Oxford Ser. (2)11 (1940) 293–303
Erdélyi A, On some functional transformations,Univ. Politec. Torino. Rend. Sem. Mat. 10 (1950–51) 217–234
Erdélyi A, Fractional integrals of generalized functions, in ‘Fractional, calculus and its applications’ (Lecture Notes in Math., Vol. 457), 151–170 New York, Springer-Verlag (1975)
Erdélyi A, Magnus W, Oberhettinger F and Tricomi F G,Higher transcendental functions, Vol. I, New York, McGraw-Hill, (1953)
Erdélyi A, Magnus W, Oberhettinger F and Tricomi F G,Tables of integral transforms, Vol. II New York McGraw-Hill, (1954)
Fox C, The G and H functions as symmetrical Fourier kernels,Trans. Am. Math. Soc. 98 (1961) 395–429
Goyal S P and Jain R M, Fractional integral operators and the generalized hypergeometric functions,Indian J. Pure Appl. Math.,18 (1987) 251–259
Gupta R, Fractional integral operators and a general class of polynomials,Indian J. Math. 32 (1990) 69–77
Kalla S L, Operators of fractional integration, Analytic Functions, Kozubunik 1979, (Lecture Notes in Math., Vol. 798), 258–280, Berlin, Springer) (1980)
Kober H, On fractional integrals and derivatives,Quart. J. Math. Oxford Ser (2)11 (1940) 193–211
Lowndes J S, A generalization of the Erdélyi-Kober operators,Proc. Edinburgh Math. Soc. Ser. (2)17 (1970) 139–148
Saxena R K, On fractional integration operators,Math. Z 96 (1967) 288–291
Saxena R K and Kumbhat R K, Integral operators involvingH-function,Indian J. Pure Appl. Math.,5 (1974) 1–6
Srivastava H M, A contour integral involving Fox’sH-function,Indian J. Math. 14 (1972) 1–6
Srivastava H M, Goyal S P and Jain R M, Fractional integral operators involving a general class of polynomials,J. Math. Anal. Appl. 148 (1990) 87–100
Srivastava H M, Gupta K C and Goyal S P, TheH-functions of one and two variables with applications, South Asian Publishers, New Delhi/Madras, 1982
Srivastava H M and Panda R, Some bilateral generating functions for a class of generalized hypergeometric polynomials,J. Reine Angew. Math. 283/284 (1976) 265–274
Srivastava H M and Singh N P, The integration of certain products of the multivariableH-function with a general class of polynomials,Rend. Circ. Mat. Palermo (2)32 (1983) 157–187
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Gupta, K.C., Soni, R.C. On composition of some general fractional integral operators. Proc. Indian Acad. Sci. (Math. Sci.) 104, 339–349 (1994). https://doi.org/10.1007/BF02863413
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DOI: https://doi.org/10.1007/BF02863413