Some generalizations of pochhammer's symbol and their associated families of hypergeometric functions and hypergeometric polynomials

Applied Mathematics and Information Sciences

Volumn 7, Issue 6, 2013, Pages 2195-2206

  Srivastava, Rekha ^a  

^a UNIVERSITY OF VICTORIA   (Canada)

Author keywords

Chu Vandermonde summation formula; Combinatorial identities; Gamma function; Generating functions; Gould's identity; Incomplete Gamma functions; Incomplete hypergeometric functions and polynomials; Incomplete Pochhammer symbols; Lagrange expansion theorem; Reduction formulas; Srivastava Buschman generating function

Indexed keywords

EID: 84884511509     PISSN: 19350090     EISSN: 23250399     Source Type: Journal    
DOI: 10.12785/amis/070609     Document Type: Article

References (51)

1. M. Abramowitz and I. A. Stegun (Editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Tenth Printing, Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., (1964); Reprinted by Dover Publications, New York, (1965) (see also [24]).
2. L. Carlitz, Some generating functions for Laguerre polynomials, Duke Math. J., 35, 825-827 (1968).
3. B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, San Francisco and London, (1977).
4. A. Çetinkaya, The incomplete second Appell hypergeometric functions, Appl. Math. Comput., 219, 8332-8337 (2013).
5. M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159, 589-602 (2004).
6. M. A. Chaudhry and S.M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman and Hall (CRC Press Company), Boca Raton, London, New York and Washington, D.C., (2001).
7. L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Company, New York, (1985).
8. W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Cambridge University Press, Cambridge, London and New York, (1935); Reprinted by Stechert- Hafner Service Agency, New York and London, (1964).
9. F. Brafman, Some generating functions of the Laguerre and Hermite polynomials, Canad. J. Math., 9, 180-187 (1957).
10. J. W. Brown, New generating functions for classical polynomials, Proc. Amer. Math. Soc., 21, 263-268 (1969).
11. T. W. Chaundy, An extension of hypergeometric functions (I), Quart. J. Math. Oxford Ser., 14, 55-78 (1943).
12. A. Erdélyi, W. Mangus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, Toronto and London, (1953).
13. A. Erdélyi, W. Mangus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. III, McGraw-Hill Book Company, New York, Toronto and London, (1955).
14. H. W. Gould, A series transformation for finding convolution identities, Duke Math. J., 28, 193-202 (1961).
15. N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Vol. 2, JohnWiley and Sons, New York, Chichester, Brisbane and Toronto, (1995).
16. P. W. Karlsson, Hypergeometric functions with integral parameter differences, J. Math. Phys., 12, 270-271 (1971),
17. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North- Holland Mathematical Studies, Vol. 204, Elsevier (North- Holland) Science Publishers, Amsterdam, London and New York, (2006).
18. K. Kobayashi, On generalized gamma functions occurring in diffraction theory, J. Phys. Soc. Japan, 61, 1501-1512 (1991).
19. K. Kobayashi, Plane wave diffraction by a strip: Exact and asymptotic solutions, J. Phys. Soc. Japan, 61, 1891-1905 (1991).
20. Y. L. Luke, Mathematical Functions and Their Approximations, Academic Press, New York, San Francisco and London, (1975).
21. W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third Enlarged Edition, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtingung der Anwendungsgebiete, Band 52, Springer-Verlag, Berlin, Heidelberg and New York, (1966).
22. A. R. Miller and H. M. Srivastava, Karlsson-Minton summation theorems for the generalized hyppergeometric series of unit argument, Integral Transforms Spec. Funct., 21, 603-612 (2010).
23. K. B. Oldham, J. Myland and J. Spanier, An Atlas of Functions. With Equator, the Atlas Function Calculator, Second Edition [With 1 CD-ROM (Windows)], Springer, Berlin, Heidelberg and New York, (2009).
24. F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (Editors), NIST Handbook of Mathematical Functions [With 1 CD-ROM (Windows, Macintosh and UNIX)], U. S. Department of Commerce, National Institute of Standards and Technology, Washington, D. C., (2010); Cambridge University Press, Cambridge, London and New York, (2010) (see also [1] above).
25. R. Panda, A note on certain reducible cases of the generalized hypergeometric function, Nederl. Akad. Wetensch. Indag. Math., 38, 41-45 (1976).
26. R. Panda, The reducible cases of certain generalized hypergeometric functions of several variables, Nederl. Akad. Wetensch. Indag. Math., 39, 469-476 (1977).
27. R. Panda, On a new class of polynomials, Glasgow Math. J.. 18, 105-108 (1977).
28. G. Pólya and G. Szegö, Problems and Theorems in Analysis, Vol. 1 (Translated from the German by D. Aeppli), Springer- Verlag, New York, Heidelberg and Berlin, (1972).
29. E. D. Rainville, Special Functions, Macmillan Company, New York, (1960); Reprinted by Chelsea publishing Company, Bronx, New York, (1971).
30. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, London and New York, (1966).
31. H. M. Srivastava, Generating functions for Jacobi and Laguerre polynomials, Proc. Amer. Math. Soc., 23, 590-595 (1969).
32. H. M. Srivastava, Generalized hypergeometric functions with integral parameter difference, Nederl. Akad. Wetensch. Indag. Math., 35, 38-40 (1973).
33. H.M. Srivastava, Some generalizations of Carlitz's theorem, Pacific J. Math., 85, 471-477 (1979).
34. H. M. Srivastava, A family of q-generating functions, Bull. Inst. Math. Acad. Sinica, 12, 327-336 (1984).
35. H. M. Srivastava, Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inform. Sci., 5, 390-444 (2011).
36. H. M. Srivastava and R. G. Buschman, Some polynomials defined by generating relations, Trans. Amer. Math. Soc., 205, 360-370 (1975); see also Addendum, Trans. Amer. Math. Soc., 226, 393-394 (1977).
37. H. M. Srivastava, M. A. Chaudhry and R. P. Agarwal, The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Transforms Spec. Funct., 23, 659-683 (2012).
38. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Acedemic Publishers, Dordrecht, Boston and London, (2001).
39. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, (2012).
40. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, (1985).
41. H. M. Srivastava and B. R. K. Kashyap, Special Functions in Queuing Theory and Related Stochastic Processes, Academic Press, New York and London, (1982).
42. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, (1984).
43. H. M. Srivastava and R. Panda, Some expansion theorems and generating relations for the H function of several complex variables. II, Comment. Math. Univ. St. Paul., 25 (2), 167-197 (1976).
44. H. M. Srivastava, R. K. Parmar and P. Chopra, A class of extended fractional derivative operators and associated generating relations involving hypergeometric functions, Axioms, 1, 238-258 (2012).
45. H. M. Srivastava, R. K. Saxena and C. Ram, A unified presentation of the Gamma-type functions occurring in diffration theory and associated probability distributions, Appl. Math. Comput., 162, 931-947 (2005).
46. R. Srivastava, New generating functions for a class of polynomials, Boll. Un. Mat. Ital. (Ser. 5), 17A, 183-186 (1980).
47. R. Srivastava, Some families of integral, trigonometric and other related inequalities, Appl. Math. Inform. Sci., 5, 342- 360 (2011).
48. R. Srivastava, Some families of combinatorial and other series identities and their applications, Appl.Math. Comput., 218, 1077-1083 (2011).
49. R. Srivastava, Some properties of a family of incomplete hypergeometric functions, Russian J. Math. Phys., 20, 121- 128 (2013).
50. R. Srivastava and N. E. Cho, Generating functions for a certain class of incomplete hypergeometric polynomials, Appl. Math. Comput., 219, 3219-3225 (2012).
51. N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, (1996).