We evaluate an integral involving an Hermite polynomial, a generalized hypergeometric series and Fox's H-function, and employ it to evaluate a double integral involving Hermite polynomials, generalized hypergeometric series and the H-function. We further utilize the integral to establish a Fourier-Hermite expansion and a double Fourier-Hermite expansion for products of generalized hypergeometric functions.
@article{bwmeta1.element.bwnjournal-article-apmv56z1p19bwm, author = {Sadhana Mishra}, title = {Integrals involving Hermite polynomials, generalized hypergeometric series and Fox's H-function, and Fourier-Hermite series for products of generalized hypergeometric functions}, journal = {Annales Polonici Mathematici}, volume = {55}, year = {1991}, pages = {19-28}, zbl = {0742.33006}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv56z1p19bwm} }
Sadhana Mishra. Integrals involving Hermite polynomials, generalized hypergeometric series and Fox's H-function, and Fourier-Hermite series for products of generalized hypergeometric functions. Annales Polonici Mathematici, Tome 55 (1991) pp. 19-28. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv56z1p19bwm/
[000] [1] S. D. Bajpai, An integral involving Fox's H-function and heat conduction, Math. Ed. (Siwan) 3 (1969), 1-4. | Zbl 0174.10602
[001] [2] S. D. Bajpai, An expansion formula for Meijer's G-function involving Hermite polynomials, Labdev J. Sci. Tech. Part A8 (1970), 9-11.
[002] [3] B. R. Bhonsle, Heat conduction and Hermite polynomials, Proc. Nat. Acad. Sci. India Sect. A 36 (1966), 359-360. | Zbl 0227.35053
[003] [4] B. L. J. Braaksma, Asymptotic expansions and analytic continuations for a class of Barnes integrals, Compositio Math. 15 (1963), 239-341. | Zbl 0129.28604
[004] [5] A. Erdélyi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York 1953. | Zbl 0051.30303
[005] [6] C. Fox, The G and H-functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395-429. | Zbl 0096.30804
[006] [7] G. K. Goyal, An integral involving H-function, Proc. Nat. Acad. Sci. India Sect. A 39 (1969), 201-203. | Zbl 0244.33011
[007] [8] K. C. Gupta and G. S. Olkha, Integrals involving the products of generalized hypergeometric functions and Fox's H-function, Univ. Nac. Tucumán Rev. Ser. A 19 (1969), 205-212. | Zbl 0185.29801
[008] [9] J. Kampé De Fériet, Heat conduction and Hermite polynomials, Bull. Calcutta Math. Soc., Golden Jubilee Commemoration Volume (1958-59), 103-104.
[009] [10] A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Lecture Notes in Math. 348, Springer, Berlin 1973. | Zbl 0272.33001
[010] [11] A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, Wiley Eastern Ltd., New Delhi 1978. | Zbl 0382.33001
[011] [12] L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan, London 1933. | Zbl 59.1111.01
[012] [13] E. D. Rainville, Special Functions, McGraw-Hill, New York 1960. | Zbl 0092.06503
[013] [14] M. Shah, On some results on the H-function involving Hermite polynomials, J. Natur. Sci. Math. 9 (1969), 223-233. | Zbl 0195.06602
[014] [15] M. Shah, Heat conduction, generalized Meijer's function and Hermite polynomials, Comment. Math. Univ. St. Paul. 19 (1970), 81-94. | Zbl 0202.05504
[015] [16] F. Singh and R. C. Varma, Application of E-operator to evaluate a definite integral and its application in heat conduction, J. Indian Math. Soc. (N.S.), 36 (1972), 325-332. | Zbl 0268.33007
[016] [17] H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-function of One and Two Variables with Applications, South Asian Publ., New Delhi 1982. | Zbl 0506.33007
[017] [18] J. Wimp and Y. L. Luke, Expansion formulae for generalized hypergeometric functions, Rend. Circ. Mat. Palermo 11 (1962), 351-366. | Zbl 0144.06902