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 
.
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
 | (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:
 | (1.2) |
where 
is a binomial coefficient and 
are complex numbers independent of n and 
is a function of 't' defined implicitly by
 | (1.3) |
subject to the condition
 | (1.4) |
Another generalization [ 2, p. 348, problem 212] related with the equation (1.2), is given as:
 | (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:
 | (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:
 | (1.7) |
with
 |
If we put 
and 
= 
in Gould's identity (1.6), we get the second modified form of Gould's identity
 | (1.8) |
with
 |
Gauss’s Multiplication Theorem
For every positive integer m, we have
 | (1.9) |
Summation identity [ 5, p, 101, Lemma (3), (2.1.6)]
 | (1.10) |
provided that series involved are absolutely convergent.
The generalized Laguerre polynomials 
are defined by
 | (1.11) |
Replacing 
by 
in equation (1.11), we get
 | (1.12) |
The Jacobi Polynomials of first kind 
[ 6, p. 254 (132.1), p. 255 (132.7)] are defined by the following equations:
 | (1.13) |
 | (1.14) |
where n is a non-negative integer.
Replacing 
by 
and 
by 
in equation (1.13), we get
 | (1.15) |
Replacing 
by 
and 
by 
in equation (1.14), we get the following result
 | (1.16) |
The generalized Rice Polynomials 
of Khandekar [ 7, p. 158, eq. (2.3)] are defined by
 | (1.17) |
 | (1.18) |
 | (1.19) |
Replacing 
by 
and 
by 
in eq.(1.17), we get
 | (1.20) |
Some useful Pochhammer’s relations
 | (1.21) |
 | (1.22) |
 | (1.23) |
 | (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
 | (1.24) |
Case II:- When 
in (1.3),we get
 |
then one of the values of 
(satisfying the condition (1.4)) is given by
 | (1.25) |
Case III:- When 
in (1.3), we get
 |
then the particular value of 
(satisfying the condition (1.4)) is given by
 | (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
 | (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
 | (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
 | (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
 | (1.30) |
Case VIII:-When 
in (1.3), we obtain
 |
then one of the values of 
(satisfying the condition (1.4)) is denoted by
 | (1.31) |
First Generating Relation:
If any values of variables and parameters leading to the results which do not make sense, are tacitly excluded, then
 | (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
 | (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
 | (2.3) |
Now applying summation identity (1.10) and then simplifying further, we get
 | (2.4) |
Now using first modified Gould's identity (1.7) and then (1.23a), we get
 | (2.5) |
Simplifying it further, we get
 | (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
 | (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
 | (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
 | (2.9) |
Now applying summation identity (1.10) in above equation then simplifying further, we get
 | (2.10) |
Now using second modified Gould’s identity (1.8), we get
 | (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).
(i). Putting 
in equation (2.1) and after simplifying, we get
 | (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
 | (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
 | (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
 | (3.4) |
Now replacing 
to 
in equation (3.4), we get
 | (3.5) |
Further taking
 |
in equation (3.5), we get
 | (3.6) |
Replacing 
by 
and 
by 
in equation (3.6), we get
 | (3.7) |
which is the known result of Srivastava [ 10; p. 233, equation (13)].
(i). Taking 
in equation (3.2), we get
 | (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
 | (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
 | (4.3) |
and taking 
in equations (1.3), (1.4) and (4.2), we have
 | (4.4) |
(ii). In equations (1.3), (1.4) and (3.2), putting 
, replacing 
by 
, 
by 
and 
by 
, we get
 | (4.5) |
where v is a function of 't', defined implicitly by
 | (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
 | (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)].
 | (4.8) |
(iii). Putting 
and replacing 
by 
in equations (1.3), (1.4) and (3.2), we get
 | (4.9) |
where
 |
Replacing 
by 
and 
by 
and then 
by 
and using the definition (1.16) of Jacobi Polynomials, we get the following result
 | (4.10) |
where
 |
Replacing 
by 
; 
by 
, we get
 | (4.11) |
where
 | (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)].
 | (4.13) |
And taking 
in equations (4.11) and (4.12), we obtain
 | (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
 | (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]:
 | (4.16) |
where
 |
(v). Putting 
and replacing 
by 
in equations (4.6) and (4.7), we get
 | (4.17) |
 |
which is the known result of Brown 13.
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
 | (5.1) |
(ii). Putting 
and 
from equation (1.25) in equation (2.1), we get
 | (5.2) |
(iii). Putting 
and 
from equation (1.26) in equation (2.1), we get
 | (5.3) |
(iv). Putting 
and 
from equation (1.27) in equation (2.1), we get
 | (5.4) |
(v). Putting 
and 
from equation (1.28) in equation (2.1), we get
 | (5.5) |
(vi). Putting 
and 
from equation (1.29) in equation (2.1), we get
 | (5.6) |
(vii). Putting 
and 
from equation (1.30) in equation (2.1), we get
 | (5.7) |
(viii). Putting 
and 
from equation (1.31) in equation (2.1), we get
 | (5.8) |
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
 | (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
 |
 | (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
 | (6.3) |
(iv). Putting 
and 
from equation (1.24) in equation (2.7), we get
 | (6.4) |
(v). Putting 
and 
from equation (1.25) in equation (2.7), we get
 |
 | (6.5) |
(vi). Putting 
and 
from equation (1.26) in equation (2.7), we get
 |
 | (6.6) |
(vii). Putting 
and 
from equation (1.27) in equation (2.7), we get
 |
 | (6.7) |
(viii). Putting 
and 
from equation (1.28) in equation (2.7), we get
 |
 | (6.8) |
(ix). Putting 
and 
from equation (1.29) in equation (2.7), we get
 |
 | (6.9) |
(x). Putting 
and 
from equation (1.30) in equation (2.7), we get
 |
 | (6.10) |
(xi). Putting 
and 
from equation (1.31) in equation (2.7), we get
 | (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.
Generalization of (2.1):
Let
 | (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
 | (7.2) |
where 
is given by
 | (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
 | (7.4) |
Applying summation identity (1.10) and then simplifying further, we get
 | (7.5) |
Now using first modified Gould's identity (1.7) with conditions (7.3), we get
 | (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
 | (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
 | (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
 | (7.9) |
Applying summation identity (1.10) and then simplifying further, we get
 | (7.10) |
Now using second modified Gould's identity (1.8) with conditions (7.3), we get
 | (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.
| [1] | Whittaker, E. T. and Watson, G. N.; A Course of Modern Analysis, Fourth ed., Cambridge Univ. Press, Cambridge, London and New York (1927). | ||
| In article | PubMed | ||
| [2] | P´olya, G. and Szeg¨o, G; Problems and Theorems in Analysis, Vol. I, (Translated from theGerman by D.Aeppli) Springer-Verlag, New York, Heidelberg and Berlin (1972). | ||
| In article | |||
| [3] | Gould, H. W.; Some generalizations of Vandermonde's Convolution, Amer. Math. Monthly, 63(1956); 84-91. | ||
| In article | View Article | ||
| [4] | Riordan, J.; Combinatorial Identities, John Wiley & Sons, New York, London and Sydney (1968). | ||
| In article | |||
| [5] | Srivastava, H. M. and Manocha, H. L.; A Treatise on Generating functions, Halsted Press (Ellis Horwood Ltd., Chichester, U. K.), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, (1984). | ||
| In article | |||
| [6] | Rainville, E. D.; Special Functions, The Macmillan, New York, (1960); Reprinted by Chelsea Publishing Company, Bronx, New York, (1971). | ||
| In article | |||
| [7] | Khandekar, P. R.; On a generalization of Rice's polynomial, I. Proc. Nat. Acad. Sci. India Sect. A, 34(1964); 157-162. | ||
| In article | |||
| [8] | Srivastava, H. M.; A Note on Certain Generating functions for the Classical Polynomi-als, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8); 63(1977), 328-333. | ||
| In article | |||
| [9] | Srivastava, H. M.; A class of Generating Functions for generalized Hypergeometric Poly-nomials (Abstract), Notices Amer. Math. Soc., 16(1969); 975 (Abstract #69T-B198). | ||
| In article | |||
| [10] | Srivastava, H. M.; A class of Generating Functions for generalized Hypergeometric Poly-nomials, J. Math. Anal. Appl., 35 (1971); 230-235. | ||
| In article | View Article | ||
| [11] | Srivastava, H. M.; Generating Functions for Jacobi and Laguerre Polynomials, Proc. Amer. Math. Soc., 23(1969); 590-595. | ||
| In article | View Article | ||
| [12] | Brown, J. W.; New Generating Functions for Classical Polynomials, Proc. Amer. Math. Soc., 21(1969); 263-268. | ||
| In article | View Article | ||
| [13] | Brown, J. W.; On Zero Type Sets of Laguerre Polynomials, Duke Math. J., 35(No.4) (1968); 821-823. | ||
| In article | View Article | ||
| [14] | Carlitz, L.; Some Generating Functions for Laguerre Polynomials, Duke Math. J., 35(1968); 825-827. | ||
| In article | View Article | ||
| [15] | Srivastava, H. M. and Singhal, J. P.; New Generating Functions for Jacobi and Related Polynomials, J. Math. Anal. Appl., 41 (1973); 748-752. | ||
| In article | View Article | ||
| [16] | Calvez, L.-C. and G'enin, R. Sur les relations entre les fonctions g´en´eratrices et les for-mules de type Rodrigues, C. R. Acad. Sci. Paris S'er. A 269(1969); 651-654. | ||
| In article | |||
| [17] | Milch, P. R.; A Probabilistic proof of a formula for Jacobi Polynomials by L. Carlitz, Proc. Cambridge Philos. Soc., 64(1968); 695-698. | ||
| In article | View Article | ||
| [18] | Joshi, C. M. and Prajapat, M. L.; A Triple integral transformation and its applications to generating functions. Boll. Un. Mat. Ital. (5), 14A (1977); 264-274. | ||
| In article | |||
| [19] | Karande, B. K. and Thakare, N. K.; Note on the generating function for generalized hy-pergeometric polynomials, Indian J. Pure Appl. Math., 6(1975); 1185-1187. | ||
| In article | |||
Published with license by Science and Education Publishing, Copyright © 2018 M.I. Qureshi and Sulakshana Bajaj
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| [1] | Whittaker, E. T. and Watson, G. N.; A Course of Modern Analysis, Fourth ed., Cambridge Univ. Press, Cambridge, London and New York (1927). | ||
| In article | PubMed | ||
| [2] | P´olya, G. and Szeg¨o, G; Problems and Theorems in Analysis, Vol. I, (Translated from theGerman by D.Aeppli) Springer-Verlag, New York, Heidelberg and Berlin (1972). | ||
| In article | |||
| [3] | Gould, H. W.; Some generalizations of Vandermonde's Convolution, Amer. Math. Monthly, 63(1956); 84-91. | ||
| In article | View Article | ||
| [4] | Riordan, J.; Combinatorial Identities, John Wiley & Sons, New York, London and Sydney (1968). | ||
| In article | |||
| [5] | Srivastava, H. M. and Manocha, H. L.; A Treatise on Generating functions, Halsted Press (Ellis Horwood Ltd., Chichester, U. K.), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, (1984). | ||
| In article | |||
| [6] | Rainville, E. D.; Special Functions, The Macmillan, New York, (1960); Reprinted by Chelsea Publishing Company, Bronx, New York, (1971). | ||
| In article | |||
| [7] | Khandekar, P. R.; On a generalization of Rice's polynomial, I. Proc. Nat. Acad. Sci. India Sect. A, 34(1964); 157-162. | ||
| In article | |||
| [8] | Srivastava, H. M.; A Note on Certain Generating functions for the Classical Polynomi-als, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8); 63(1977), 328-333. | ||
| In article | |||
| [9] | Srivastava, H. M.; A class of Generating Functions for generalized Hypergeometric Poly-nomials (Abstract), Notices Amer. Math. Soc., 16(1969); 975 (Abstract #69T-B198). | ||
| In article | |||
| [10] | Srivastava, H. M.; A class of Generating Functions for generalized Hypergeometric Poly-nomials, J. Math. Anal. Appl., 35 (1971); 230-235. | ||
| In article | View Article | ||
| [11] | Srivastava, H. M.; Generating Functions for Jacobi and Laguerre Polynomials, Proc. Amer. Math. Soc., 23(1969); 590-595. | ||
| In article | View Article | ||
| [12] | Brown, J. W.; New Generating Functions for Classical Polynomials, Proc. Amer. Math. Soc., 21(1969); 263-268. | ||
| In article | View Article | ||
| [13] | Brown, J. W.; On Zero Type Sets of Laguerre Polynomials, Duke Math. J., 35(No.4) (1968); 821-823. | ||
| In article | View Article | ||
| [14] | Carlitz, L.; Some Generating Functions for Laguerre Polynomials, Duke Math. J., 35(1968); 825-827. | ||
| In article | View Article | ||
| [15] | Srivastava, H. M. and Singhal, J. P.; New Generating Functions for Jacobi and Related Polynomials, J. Math. Anal. Appl., 41 (1973); 748-752. | ||
| In article | View Article | ||
| [16] | Calvez, L.-C. and G'enin, R. Sur les relations entre les fonctions g´en´eratrices et les for-mules de type Rodrigues, C. R. Acad. Sci. Paris S'er. A 269(1969); 651-654. | ||
| In article | |||
| [17] | Milch, P. R.; A Probabilistic proof of a formula for Jacobi Polynomials by L. Carlitz, Proc. Cambridge Philos. Soc., 64(1968); 695-698. | ||
| In article | View Article | ||
| [18] | Joshi, C. M. and Prajapat, M. L.; A Triple integral transformation and its applications to generating functions. Boll. Un. Mat. Ital. (5), 14A (1977); 264-274. | ||
| In article | |||
| [19] | Karande, B. K. and Thakare, N. K.; Note on the generating function for generalized hy-pergeometric polynomials, Indian J. Pure Appl. Math., 6(1975); 1185-1187. | ||
| In article | |||
