LINEAR AND BILINEAR GENERATING FUNCTIONS OF HEAT TYPE POLYNOMIALS SUGGESTED BYJACOBI POLYNOMIALS

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LINEAR AND BILINEAR GENERATING FUNCTIONS

OF HEAT TYPE POLYNOMIALS SUGGESTED

BYJACOBI POLYNOMIALS

Manoj Singh

, Aftab Alam

Department of Mathematics, Faculty of Science,

Jazan University, Jazan, Saudi Arabia.

2

Department of Mathematics, SRM University,

NCR Campus, Ghaziabad, India.

ABSTRACT

The object of the manuscript is to obtain several types of linear and bilinear generating functions for the heat type polynomials suggested by Jacobi polynomials.

Mathematics Subject Classification(2010

): Primary 42C05, Secondary 33C45

.

Keywords: Jacobi polynomials, Appell’s functions, Kamp de F riet’s double hypergeometric

function, Lauricella’s functions, Gauss hypergeometric function.

I. INTRODUCTION

The present paper deals with an extension of earlier work (seeM.A. Khan, et al.[1]).For the generalized heat polynomials defined by (see Haimo[2], p. 736, eq. (2.1)).

The heat type polynomials suggested by Jacobi polynomials are denoted by, and defined as

In the form similar to one given by H.M.Srivastava [3] for Jacobi polynomials (1.3) can be written as

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By means of the relation,

which one can obtain by replacing by and by in our earlier work ([1], eq.(2.2)).

another form of (1.3) is given by

In finite series form (1.3), (1.6), and (1.8) can be written as,

Next we rewrite (1.9), (1.10) and (1.11) by reversing the order of summation in the form

Some of the definition and notations used in this paper are as follows:

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Further, Kamp de F riet’s type general double hypergeometric series (seeH.M. Srivastava,et al. [6]) is defined as

Similarly, a general triple hypergeometric series (see H.M. Srivastava [7], pp.428) is defined as

where for convenience

II. LINEAR GENERATING FUNCTIONS:

By using the relation (1.3), we obtain the certain generating function for is as follows:

where,

the limiting case of (2.1) when yields

and

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and make use of Pfaff- Kummer transformation (see [8], pp.64)

to each term of the series and replacing on both sides of the resulting equation by , (2.4) thus leads us to the

generating function (2.3).

A further limiting case of the generating function (2.3) when (or equivalently, of the generating

function (2.1) when min ) would similarly yields the result (see [1], eq.(2.1)),

where is given by (2.2).

Relation (2.6) follows also when we set

in the generating function (2.3) or when we set

in the generating function (2.1). Also, the generating function (2.3) can be deduced from (2.1) by setting

Alternative Derivation of the Generating Function(2.1):

By using one of many of special cases of his hypergeometric function (see[9], pp.76, eq.(3.1)), we thus obtain

the generating function,

Setting,

the result (2.7) at once yields,

The left-hand sides of (2.1) and (2.9) are identical. In order to show their right-hand sides are also identical, let

denote the second member of the generating function (2.9). Then it follows from the definition (1.19) and

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that,

Now replace the summation index in (2.11) by and rearrange the resulting triple sum to the form:

By applying Chu-Vandermonde theorem (see [10], pp.13) to sum the Gauss hypergeometric series with

argument 1, we obtain

Finally, by appealing to Whipple’s transformation (see [10], pp.97, eq.4(iv)) and interpreting the double series

by means of the definition (1.19), (2.12) leads the second member of generating function (2.1), which is further

special case of the result (2.7).

Now we find some more extended linear generating function for the polynomials as follow

where denotes the fourth type of Appell’s hypergeometric function of two variables defined by (1.18).

The generating relation (2.13) is an immediate consequence of the definition (1.6) and (1.18), and the

familiar Gaussian hypergeometric transformation(see[11], eq.21, pp.33).

An interesting special case of the generating function (2.13) occurs when we set and and

appealing the hypergeometric reduction formula,

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Another extended generating function are obtained by replacing by and making use of Binomial

theorem on the left-hand side, and expanding the Gaussian function on right-hand side in the generating

function(seeM.A. Khan et al.[1], eq.(2.25)),

which readily gives the relation in the elegant form

Now equate the coefficient of in (2.16) and on replacing and by we thus

arrive to the generating function,

Similarly, replace by in the generating function (see[1], eq.(2.26)),

and make use of Binomial theorem, we obtain

By using in (2.19) and in conjunction with the hypergeometric reduction formula (see,

Erd lyi et al. [9], vol.I, pp.238, eq.(1)).

a new generating relation is obtained as follows:

If we rewrite (1.14) as

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Similarly, from (1.12) we obtain

III. BILINEAR GENERATING FUNCTIONS

:

The polynomials admits the following generating functions involving the sets

and

where is a non-negative integer,and the parameter are independent on .

In view of the definition (1.10) and the generating function (2.13) would readily yields the generating

function

where denote the Lauricella’s function defined by([11], eq.1.7(3)) with

In (3.1) set and appealing the hypergeometric reduction formula

(see,B.L.Sharma[14], pp.716, eq.(2.4))

where,

leads the generating relation

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If we rewrite (1.14) as

which in conjunction with (2.17) would leads at once the generating function

where,

The second member of (3.5) can indeed be written in terms of Srivastava triple hypergeometric series

defined by (1.20), and we thus obtain the alternative form of the bilinear generating function (3.5)

as,

Another interesting special case of (3.6) would occurs, when we set , along with([5], pp.35,

eq.10).

and the power series identity(H.M. Srivastava, et al.[11], 1.6(2)). We thus obtain the generating function

where,

REFERENCES:

[1] M.A. KHAN, A.H. KHAN AND M. SINGH,A note on heat type polynomials suggested by Jacobi

polynomials, Thai Journal of Mathematics,No. 1, Vol. 10(2012),pp. 113-122.

[2] D.T. HAIMO,Expansion in terms of generalized heat polynomials and their Appell transform,J. Math. Mech., 15,(1966), 735-758.

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[4] E.D. RAINVILLE,Special Functions (MacMillan, New York (1960), Reprinted by Chelsea Publishing Company, Bronx, New York 1971).

[5] P.APPELL, J. KAMP DE F RIET,Fonctions hyp rgeom triques et hyperspheriques (Polyn mes d’

Hermite Gauthier-villars, Paris, 1926).

[6] H.M. SRIVASTAVA AND P.W. KARLSSON,Multiple Gaussian hypergeometric series (Halsted press

(Ellis Horwood Limited, Chichester) , John wiley and sons, New York, Chichester, 1985).

[7] H.M. SRIVASTAVA,Generalized Neumann expansion involving hypergeometric functions, Proc. Camb.

Phil. Soc., 63, 425-429, (1967).

[8] A. ERDELYI, W. MAGNUS, F. OBERHETTINGER AND F.G. TRICOMI,Higher Transcendental

Functions (I. McGraw-Hill, New York Toronto and London,1953).

[9] H.M. SRIVASTAVA,Certain formulas associated with the generalized Rice polynomials, II, Ann. Polon. Math., 27, (1972), 73-83.

[10] W.N. BAILEY,Generalized hypergeometric series (Cambridge Math., Tract No.32, Cambridge Univ.

Press, Cambridge, London and New York, (1935). Reprinted by Stechert-Hafner service Agency, New

York and London, 1964).

[11] 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, Toronto1985).

[12] B.L. SHARMA,Integrals involving hypergeometric functions of two variables, Proc. Nat. Acad. Sci. India. Sec., A-36,(1966), 713-718.

ADDRESS FOR CORRESPONDENCE

Manoj Singh

Department Mathematics, Faculty of Science,

Jazan University, Jazan, Saudi Arabia