International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013
ISSN 2250-3153
1
An integral involving general class of polynomials with Ifunction
Jyoti Mishra* and Neelam Pandey**
Department of Mathematics, Gyan Ganga College of Technology, Jabalpur*
Model Science College, Rewa**
Abstract- In the present paper we derive an integral involving general class of polynomials whose integrand contains product of Ifunction and a general class of polynomiyals.The integral evaluated is quite general in nature and we can derive from them by a large
number of integrals involving simpler functions as their particular cases.
Index Terms- I-function, general class of polynomials, Fox H-function.
I. INTRODUCTION
T
he I-function of one variable introduced by Saxena [1982],will be represented and defined in the following manner:
(1.1)
Where
is a complex variable and
in which log |z|represent the natural logarithm of |z| and arg |z| is not necessarily the principal value. An empty product is interpreted
as unity.
z is not equal to zero and an empty product is interpreted as unity;
satisfying
are non -negative integer
: forall
and
are assumed to be positive quantities for standardization purpose.
Also
A general class of polynomial
following manner:
and
complex numbers.
occurring in the paper was introduced by Srivastava [4] is defined and represented in the
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International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013
ISSN 2250-3153
2
V = 0, 1, 2, ..... where U is an arbitrary positive integer and coefficients
are arbitrary constant, real or complex.
II. MAIN RESULT
(2.1)
Where
We shall require the following elementary integral for the evaluation of our main integral
Proof :To obtain the result (2.1) , we first express I-function involved in its left -hand side in terms of contour integral using eq.(1.1)
and general class of polynomiyals
given by in eq.(1.2) and Interchanging the order of integration and summation (which is
permissible under the conditions stated) with (2.1) and evaluating the y-integral with the help of the result from given by equation
(2.2) so we arrive at the main result (2.1) after a little simplification:
5 Particular case : It may be noted that on account of generalized nature of the I -function,several new intersting result can be drived
with proper choice parameters. Hence formulae estaiblish in this paper are of general character.
Two particular cases should be mentoined here:
(i) In equation (2.1) setting r=1 and using (1.2) we obtained the result integrals involving Fox H-function.
(ii) In equation (2.1) by taking r=1 and
so we obtained the result integrals involving Meijer’s G-function.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Raiville , E. D.,Special Function , Macmillan and Co. N. Y. 1967
Oberhettinger F, Tables of Mellin transforms (Berlin , Heidelberg , New York: Springer-Verlag)(1974)p.22.
H.M.Srivastava and M. Garg ,Some integrals involving a general class of polynomials and the multivariable H-function,Rev.Rouinaine Phys.,32(1987), 685−692.
H.M.Srivastava,A contour integral involving fox’s H-function , Indian J. Math. 14(1972), 1 − 6
Mridula Garg and Shweta Mittal, On a new unified integral, Indian Acad. Sci. (Math. Sci.) Vol.114, No.2, 2004,pp.99 − 101
Shrivastava, H. M. ,Gupta ,K.C. and Goyal, s.p. The H- function of one and two Variables with Applications , South Asian Publishers, New Delhi 1982.
Shrivastava, H. M.,Manocha HC (1984). A Treatise on generating functions, Ellis Horwood Ltd . Chickester,John Wiley and Sons , Newyork.
ical style—Submitted for publication),” IEEE J. Quantum Electron., submitted for publication.
AUTHORS
First Author – Jyoti Mishra, Department of Mathematics, Gyan Ganga College of Technology, Jabalpur
Email id - [email protected].
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International Journal of Scientific and Research Publications, Volume 3, Issue 1, January 2013
ISSN 2250-3153
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Second Author – Neelam Pandey, Model Science College, Rewa Email id - [email protected]
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