On Certain Unified Fractional Integrals Pertaining to Product of Srivastava's Polynomials and N-Function

D.L. Suthar, G.V. Reddy, Biniyam Shimelis "On Certain Unified Fractional Integrals Pertaining to Product of Srivastava's Polynomials and N-Function", *International Journal of Mathematics Trends and Technology (IJMTT). *V47(1):66-73 July 2017. ISSN:2231-5373. www.ijmttjournal.org. Published by Seventh Sense Research Group.

**Abstract**

This paper deals with the evaluate of the fractional integrals involving Saigo
operators of the product of the Srivastava's polynomials and the N-function containing
the factor x(xk + ck) in its argument. Some interesting special cases are derived. The
results given by Chaurasia and Gupta [1] and Saigo and Raina [11] follow as special cases
of the results proved in this paper.

**Reference**

(1) V.B.L. Chaurasia and N. Gupta, General fractional integral operators, General
class of polynomials and Fox's H-function. Sochow J. Math., 25 (1999), 333-339.

(2) V.B.L. Chaurasia and Y. Singh, New generalization of integral equations of fredholm
type using Aleph function, Int. J. of Modern Math. Sci., 9(3), (2014),
208{220.

(3) C. Fox: The G and H-functions as symmetrical Fourier kernels, Trans. Amer.
Math. Soc., 98 (1961), 395-429.

(4) A.A. Kilbas and M. Saigo, H-Transforms: Theory and Application, Chapman &
Hall/CRC Press, Boca Raton, London, New York, (2004).

(5) A.M. Mathai and R. K. Saxena, The H- functions with Application in Statistics
and Other Disciplines, Halsted press, John Wiley and Sons, New York, London
and Sydney, (1978).

(6) A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-function: Theory and Applica-
tions, Springer, New York, (2010).

(7) A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series, More
Special Functions, Vol. 3, Gordon and Breach Science Publishers, New York,
(1990).

(8) S.D. Purohit, D.L. Suthar and S.L. Kalla, Some results on fractional calculus
operators associated with the M-function. Hadronic J., 33(3),(2010), 225-235

(9) E.D. Rainville, Special function, Chelsea publishing Co. Bronx, New York, (1971).

(10) M. Saigo, A remark on integral operators involving the Gauss hypergeometric
functions, Math. Rep. College General Ed. Kyushu Univ., 11, (1978), 135-143.

(11) M. Saigo and R. K. Raina, Fractional calculus operators associated with a general
class of polynomials, Fukuoka Univ. Sci. Reports, 18(1), (1988), 15-22.

(12) S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives:
Theory and Applications; Translated from the Russian: Integrals and Derivatives
of Fractional Order and Some of Their Applications (\Nauka i Tekhnika", Minsk,
1987); Gordon and Breach Science Publishers: Reading, UK, (1993).

(13) V.P. Saxena, Formal solution of certain new pair of dual integral equations involving
H-functions. Proc. Nat. Acad. Sci. India Sect., A 52, (1982), 366{375.

(14) R.K. Saxena, and T.K. Pogfany, Mathieu-type series for the N-function occurring
in Fokker-Planck equation. Eur. J. Pure Appl. Math., 3(6), (2010), 980-988.

(15) R.K. Saxena, and T.K. Pogfany, On fractional integration formulae for Aleph functions.
Appl. Math. Comput., 218, (2011), 985-990.

(16) R.K. Saxena, J. Ram and S.L. Kalla, Unified fractional integral formulas for the
generalized H-function, Rev. Acad. Canar. Cienc. XIV, (2002), 97- 109.

(17) R.K. Saxena, J. Ram and D.L. Suthar, Integral formulas for the H-function generalized
fractional calculus II. South East Asian J. Math. & Math. Sc., 5(2),
(2007), 23-31.

(18) H.M. Srivastava: A contour integral involving Fox's H-function, Indian J. Math.,
14, (1972), 1-6.

(19) H.M. Srivastava, K.C. Gupta and S.P. Goyal, The H-Functions of One and Two
Variables with Applications, South Asian Publishers, New Delhi, (1982).

(20) N. Sudland, B. Baumann and T.F. Nannenmacher, Open problem: Who knows
about the N-function?. Appl. Anal., 1(4), (1998), 401- 402.

(21) N. Sudland, B. Baumann and T.F. Nannenmacher, Fractional driftless Fokker-
Planck equation with power law diffusion coeffcients, in V.G. Gangha, E.W. Mayr,
W.G. Vorozhtsov (Eds.), Computer Algebra in Scientific Computing (CASC Konstanz
2001), Springer, Berlin, (2001), 513{525.

**Keywords**

N-function, Srivastava's polynomial, Saigo operators, generalized hypergeometric
function.