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Some Hypergeometric Generating Relations Motivated by the Work of Srivastava and Their Generalizations

M.I. Qureshi, Sulakshana Bajaj*

Department of Applied Sciences and Humanities, Modi Institute of Technology, Rawatbhata Road, Kota, Rajasthan 324010, India

Abstract: In the present paper, we have obtained hypergeometric generating relations associated with two hypergeometric polynomials of one variable and with their independent demonstrations via Gould's identity. As applications, some well known and new generating relations are deduced. Using bounded sequences, further generalizations of two main hypergeometric generating relations have also been given for two generalized polynomials and .

Keywords: Jacobi Polynomials, generalized Laguerre polynomial, generalized Rice polynomial of Khandekar, Gould's identity.

Received June 19, 2018; Revised August 22, 2018; Accepted October 08, 2018

Citation: M.I. Qureshi, Sulakshana Bajaj, Some Hypergeometric Generating Relations Motivated by the Work of Srivastava and Their Generalizations. Turkish Journal of Analysis and Number Theory, 6 (5). 136-147. January 2018. doi:10.12691/tjant-6-5-3

1. Introduction and Preliminaries

Throughout in the present paper, we use the following standard notations:

Here, as usual, denotes the set of integers, denotes the set of real numbers, denotes the set of positive real numbers and denotes the set of complex numbers.

The Pochhammer symbol (or the shifted factorial) is defined, in terms of the familiar Gamma function, by

[picture] | (1.1)

it is being understood conventionally that and assumed tacitly that the Gamma quotient exists.

Some useful consequences of Lagrange’s expansion [[1], p.133; see also [2], p.146, problem 207] include the following generalization [[2], p. 349, problem 216] of the familiar binomial expansion:

[picture] | (1.2)

where is a binomial coefficient and are complex numbers independent of n and is a function of 't' defined implicitly by

[picture] | (1.3)

subject to the condition

[picture] | (1.4)

Another generalization [[2], p. 348, problem 212] related with the equation (1.2), is given as:

[picture] | (1.5)

where is defined by the equations (1.3) and (1.4).

When both results (1.2) and (1.5) reduce immediately to the binomial expansion.

Gould [[3], p.90; see also [4], p.169] gave the following identity:

[picture] | (1.6)

where are complex parameters independent of n and is given by the equations (1.3) and (1.4).

If we put and = in Gould's identity (1.6), we get the first modified form of Gould's identity:

[picture] | (1.7)

with

If we put and = in Gould's identity (1.6), we get the second modified form of Gould's identity

[picture] | (1.8)

with

Gauss’s Multiplication Theorem

For every positive integer m, we have

[picture] | (1.9)

Summation identity [[5], p, 101, Lemma (3), (2.1.6)]

[picture] | (1.10)

provided that series involved are absolutely convergent.

The generalized Laguerre polynomials are defined by

[picture] | (1.11)

Replacing by in equation (1.11), we get

[picture] | (1.12)

The Jacobi Polynomials of first kind [[6], p. 254 (132.1), p. 255 (132.7)] are defined by the following equations:

[picture] | (1.13)

[picture] | (1.14)

where n is a non-negative integer.

Replacing by and by in equation (1.13), we get

[picture] | (1.15)

Replacing by and by in equation (1.14), we get the following result

[picture] | (1.16)

The generalized Rice Polynomials of Khandekar [[7], p. 158, eq. (2.3)] are defined by

[picture] | (1.17)

[picture] | (1.18)

[picture] | (1.19)

Replacing by and by in eq.(1.17), we get

[picture] | (1.20)

Some useful Pochhammer’s relations

[picture] | (1.21)

[picture] | (1.22)

[picture] | (1.23)

[picture] | (1.23a)

where

Now we shall discuss some special cases of the implicit functions defined by equation (1.3) subject to the condition (1.4). Using Mathematica 9.0, we can find the roots of resulting cubic equation in for different values of in equation (1.3).

Case I:- When in (1.3), then particular value of (satisfying the condition (1.4)) is denoted by

[picture] | (1.24)

Case II:- When in (1.3),we get

then one of the values of (satisfying the condition (1.4)) is given by

[picture] | (1.25)

Case III:- When in (1.3), we get

then the particular value of (satisfying the condition (1.4)) is given by

[picture] | (1.26)

Case IV:- When in (1.3), we get

then one of the roots (satisfying the condition (1.4)) of above equation is given by

[picture] | (1.27)

Case V:- When in (1.3), we get

then one of the roots (satisfying the condition (1.4)) of above equation is given by

[picture] | (1.28)

Case VI:- When in (1.3), we get

then one of the roots (satisfying the condition (1.4)) of above equation is given by

[picture] | (1.29)

where

Case VII:- When in (1.3),we obtain

then one of the values of (satisfying the condition (1.4)) is denoted by

[picture] | (1.30)

Case VIII:-When in (1.3), we obtain

then one of the values of (satisfying the condition (1.4)) is denoted by

[picture] | (1.31)

2. Main Generating Relations

First Generating Relation:

If any values of variables and parameters leading to the results which do not make sense, are tacitly excluded, then

[picture] | (2.1)

where,

provided that involved series on both sides are absolutely convergent.

Here Srivastava's generalized hypergeometric polynomials [[5], p. 360, eq. (7.3.3.); see also [8], pp. 331-332] are given by

[picture] | (2.2)

where and are complex parameters independent of ‘n’ and abbreviates the array of m number of parameters given by

Independent Demonstration:

Using the definition (2.2) of and then the power series form of in left hand side of equation (2.1), we get

Using Gauss's multiplication theorem (1.9) in above equation, we get

[picture] | (2.3)

Now applying summation identity (1.10) and then simplifying further, we get

[picture] | (2.4)

Now using first modified Gould's identity (1.7) and then (1.23a), we get

[picture] | (2.5)

Simplifying it further, we get

[picture] | (2.6)

After solving it further, we get the result (2.1) in the form of sum of two generalized hypergeometric functions of one variable.

Second Generating Relation:

If any values of variables and parameters leading to the results which do not make sense, are tacitly excluded, then

[picture] | (2.7)

where

provided that involved series on both sides are absolutely convergent.

Here we define new generalized hypergeometric polynomials known as “Pathan’s generalized hypergeometric polynomials of one variable”, given by

[picture] | (2.8)

where and are complex parameters independent of ‘n’ and abbreviates the array of m number of parameters given by

Independent Demonstration:

Using the definition (2.8) of and then the power series form of in left hand side of equation (2.7), using Gauss's multiplication theorem (1.9) and result (1.21), we get

[picture] | (2.9)

Now applying summation identity (1.10) in above equation then simplifying further, we get

[picture] | (2.10)

Now using second modified Gould’s identity (1.8), we get

[picture] | (2.11)

Now using equation (1.22) in above equation and summing it up into hypergeometric form further, we get the desired result (2.7).

3. Known Applications of Generating Relation (2.1)

(i). Putting in equation (2.1) and after simplifying, we get

[picture] | (3.1)

which is the result of Srivastava [[9], p. 975; [10], p. 233, eq. (12)]. Here being given by equations (1.3), (1.4) and is given by equation (2.2).

(ii). Putting and in equation (2.1) and using the definition of and after simplification, we get

[picture] | (3.2)

which is the result of Srivastava [[11], p. 591, eq. (9); see also 7, p. 1186]. is given by equations (1.3) and (1.4).

(iii). Putting and in equation (2.1), using the definition (2.2) of and replacing by , we get

[picture] | (3.3)

where

which is the known result of Brown [[12], p. 264, eq. (7)] and is Brown's generalized hypergeometric polynomial [[5], p. 358, eq. (7.2.4)].

(iv). Putting and replacing by in equation (3.1) and after simplifying, we get

[picture] | (3.4)

Now replacing to in equation (3.4), we get

[picture] | (3.5)

Further taking

in equation (3.5), we get

[picture] | (3.6)

Replacing by and by in equation (3.6), we get

[picture] | (3.7)

which is the known result of Srivastava [[10]; p. 233, equation (13)].

4. Some Special Cases of Generating Relation (3.2)

(i). Taking in equation (3.2), we get

[picture] | (4.1)

which is the known result of Srivastava [[11], p. 591, eq.(7)] subject to the conditions (1.3) and (1.4).

Now using the definition of generalized Laguerre polynomials (1.12) and solving, we get

[picture] | (4.2)

where is given by equations (1.3) and (1.4). It is the known result of Brown [[13], p. 822] and Carlitz [[14], p. 826; see also [11], p. 590, eq. (4)]. The generating relation (4.2) is a unification and generalization of the following two generating relations (4.3) and (4.4):

Taking in equations (1.3), (1.4) and (4.2), we have

[picture] | (4.3)

and taking in equations (1.3), (1.4) and (4.2), we have

[picture] | (4.4)

(ii). In equations (1.3), (1.4) and (3.2), putting , replacing by , by and by , we get

[picture] | (4.5)

where v is a function of 't', defined implicitly by

[picture] | (4.6)

It is the known Generating relation of Srivastava [[11], p. 591, eq. (8)]. Putting in equation (4.5) and using the definition (1.15) of Jacobi polynomials, we get

[picture] | (4.7)

where is given by equation (4.6). This is the known result of Srivastava [[11], p. 594, eq. (22); [15], p. 748; see also [16], p. A654].

Taking in equations (4.6) and (4.7), we get the following generating relation of E. Feldheim, recorded in the monograph of Srivastava-Manocha [[5], p. 90, Q. 15 (second equation); see also [11], p. 594, eq. (23)].

[picture] | (4.8)

(iii). Putting and replacing by in equations (1.3), (1.4) and (3.2), we get

[picture] | (4.9)

where

Replacing by and by and then by and using the definition (1.16) of Jacobi Polynomials, we get the following result

[picture] | (4.10)

where

Replacing by ; by , we get

[picture] | (4.11)

where

[picture] | (4.12)

which is known result of Srivastava [[11], p. 593, eq. (16); see also [15], p. 748].

Taking in equations (4.6) and (4.7); (4.11) and (4.12), we obtain another result of E. Feldheim, recorded in the monograph of Srivastava-Manocha [[5], p. 90, Q. 15 (first equation); see also [11], p. 593, eq. (19)].

[picture] | (4.13)

And taking in equations (4.11) and (4.12), we obtain

[picture] | (4.14)

which is the result of Milch [17] and also recorded in the monograph of Srivastava-Manocha [[5], p. 82, (1.11.2)].

(iv). Putting and replacing by in equations (1.3), (1.4) and (3.2), we get

[picture] | (4.15)

Now using the definition (1.20) of Generalized Rice polynomials of Khandekar in above equation (4.15), we get a well known result of Joshi and Prajapat [[18], p. 272]:

[picture] | (4.16)

where

(v). Putting and replacing by in equations (4.6) and (4.7), we get

[picture] | (4.17)

which is the known result of Brown [13].

5. New Applications of Generating Relation (2.1)

The results from (5.1) to (5.8) are believed to be new in author’s knowledge and are not found in the literature of generating relations.

(i). Putting and from equation (1.24) in equation (2.1), we get

[picture] | (5.1)

(ii). Putting and from equation (1.25) in equation (2.1), we get

[picture] | (5.2)

(iii). Putting and from equation (1.26) in equation (2.1), we get

[picture] | (5.3)

(iv). Putting and from equation (1.27) in equation (2.1), we get

[picture] | (5.4)

(v). Putting and from equation (1.28) in equation (2.1), we get

[picture] | (5.5)

(vi). Putting and from equation (1.29) in equation (2.1), we get

[picture] | (5.6)

(vii). Putting and from equation (1.30) in equation (2.1), we get

[picture] | (5.7)

(viii). Putting and from equation (1.31) in equation (2.1), we get

[picture] | (5.8)

6. New Applications of Generating Relation (2.7)

The following Generating relations of this section are new in the author’s knowledge and are not available in the literature of Generating relations.

(i). Putting in equation (2.7) and after simplifying, we get

[picture] | (6.1)

where is given by equations (1.3) and (1.4).

(ii). Putting in equation (2.7) and using the definition of , we get

[picture] | (6.2)

where is given by equations (1.3) and (1.4).

(iii). Putting in equation (2.7), using the definition of and replacing by from equation (1.28), we get

[picture] | (6.3)

(iv). Putting and from equation (1.24) in equation (2.7), we get

[picture] | (6.4)

(v). Putting and from equation (1.25) in equation (2.7), we get

[picture] | (6.5)

(vi). Putting and from equation (1.26) in equation (2.7), we get

[picture] | (6.6)

(vii). Putting and from equation (1.27) in equation (2.7), we get

[picture] | (6.7)

(viii). Putting and from equation (1.28) in equation (2.7), we get

[picture] | (6.8)

(ix). Putting and from equation (1.29) in equation (2.7), we get

[picture] | (6.9)

(x). Putting and from equation (1.30) in equation (2.7), we get

[picture] | (6.10)

(xi). Putting and from equation (1.31) in equation (2.7), we get

[picture] | (6.11)

Making suitable adjustments of parameters and variables in all generating relations of sections 5 and 6, we can also obtain a number of new generating relations involving restricted generalized Laguerre polynomials, restricted Jacobi polynomials, restricted generalized Rice polynomials of Khandekar and other orthogonal polynomials.

7. Further Generalizations of Generating Relations (2.1) and (2.7)

Generalization of (2.1):

Let

[picture] | (7.1)

where are complex parameters independent of `n'; m is an arbitrary positive integer and is a bounded sequence of arbitrary real and complex numbers such that Then

[picture] | (7.2)

where is given by

[picture] | (7.3)

provided that each of the series involved is absolutely convergent.

Independent Demonstration:

Using the definition (7.1) of in left hand side of equation (7.2), we get

[picture] | (7.4)

Applying summation identity (1.10) and then simplifying further, we get

[picture] | (7.5)

Now using first modified Gould's identity (1.7) with conditions (7.3), we get

[picture] | (7.6)

Changing the summation index from r to n and after solving it further,we get the general result (7.2) corresponding to our first generating relation (2.1) subject to the conditions (7.3).

Generalization of (2.7):

Let

[picture] | (7.7)

where are complex parameters independent of `n'; m is an arbitrary positive integer and is a bounded sequence of arbitrary real and complex numbers such that Then

[picture] | (7.8)

where is given by

provided that each of the series involved is absolutely convergent.

Independent Demonstration:

Using the definition (7.7) of in left hand side of equation (7.8), we get

[picture] | (7.9)

Applying summation identity (1.10) and then simplifying further, we get

[picture] | (7.10)

Now using second modified Gould's identity (1.8) with conditions (7.3), we get

[picture] | (7.11)

Changing the summation index from to and after solving it further, we get the general result (7.8) corresponding to our second generating relation (2.7) subject to the conditions (7.3).

In the definitions of generalized polynomials given by and , putting

we obtain Srivastava's generalized hypergeometric polynomials of one variable and Pathan's generalized hypergeometric polynomials of one variable respectively.

References

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[3]   | Gould, H. W.; Some generalizations of Vandermonde's Convolution, Amer. Math. Monthly, 63(1956); 84-91.
[4]   | Riordan, J.; Combinatorial Identities, John Wiley & Sons, New York, London and Sydney (1968).
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