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ABSTRACT
The author presents a comprehensive analysis of some bilinear generating relations pertaining the q-polynomial class and the q-analogue of the I-function. The related conclusions for q-polynomials of q-Laguerre, q-Jacobi polynomials, and so on are very simple to derive as particular or limiting examples of the generating functions provided here. It is also presented various generalized integrals related to q-polynomials and Iq-function. Furthermore, we obtain several results as extensions of these evaluations.
1. Introduction and preliminaries
Extensions of the ordinary calculus are called the q-calculus. Recent applications of the theory of q-calculus operators include q-transform analysis, optimum control problems, solving the q-difference and q-integral equations, and ordinary fractional calculus. Analyzing bilinear generating relations within the context of q-polynomial classes involves examining relationships between different generating functions that arise in combinatorial problems.
Generating functions play a crucial role in enumerative combinatorics. They encode sequences of numbers (often counting the number of objects of a certain type) into a formal power series. Bilinear generating relations often connect different generating functions, providing insights into their combinatorial interpretations and properties. Within the field of generating q-theory, several researchers are investigating applications of these relations in solving combinatorial problems, deriving identities, and understanding the behavior of special functions in various contexts. Among them are Al-Omari et al. (Citation2021), Purohit et al. (Citation2012), Srivastava et al. (Citation2010), Meena et al. (Citation2022), Ahuja & Çetinkaya (Citation2021), Vyas et al. (Citation2021) and Araci (Citation2021), presented a unique method of generating function and examined its characteristic. Proving bilinear generating relations for the q-analogue of the I-function typically involves manipulating q-series expansions, exploiting properties of q-Pochhammer symbols, contour integration techniques, and using combinatorial arguments.
The q-Pochhammer symbol, denoted as , is another essential q-polynomial related to q-factorials. Bilinear generating relations involving q-Pochhammer symbols often lead to interesting identities and transformations.
The q-shifted factorial (see Gasper & Rahman, Citation2004) for is defined as
as well as its natural continuation
Since is a convergent infinite product, the expression (Equation1.1
(1.1)
(1.1) ) is still valid.
According to Ernst (Citation2003), the power function has the following q-analogue:
where the description of the q-binomial coefficient is
It satisfies
From Ernst (Citation2003) (p. 502, equation (3.18)), given a restricted sequence of real or complex numbers and letting
be a power series in g, we may calculate the following:
According to Gasper & Rahman (Citation2004), the q-gamma function is described as
and it satisfies
According to Srivastava & Agarwal (Citation1989), family of q-polynomials ()
described in terms of a bounded complex sequence
as
where γ is an integer with a positive sign.
The family of q-polynomial provides numerous well-known q-polynomials as their particular cases when the sequence
is chosen appropriately. These include Wall polynomials, q-Konhauser polynomials, q-Jacobi polynomials, q-Laguerre polynomials, and a number of others.
Now, as indicated in Saxena & Kumar (Citation1995), q-analogue of I-function (called Iq-function) in terms of Mellin–Barnes type basic contour integral is defined as
where
and
Also, ,
is finite;
and
are complex numbers;
are real and positive.
The contour C extends from to
, with poles of
to the left and those
to the right of C. The integral converges for large values of
, if
on the contour C, which means if
. It is possible to see that the integration contour C can be substituted by other appropriately indented contours parallel to the imaginary axis.
It is intriguing to note that for r = 1, ; thus, (Equation1.11
(1.11)
(1.11) ) provides q-analogue of H-function attributed to Saxena et al. (Citation1983), precisely
where
In addition, and
are all positive integers and
. The contour C is a line parallel to
with any required indentations placed so that the poles of both
and
are to the left and right of C, respectively. The integral converges for large values of
, if
on the contour C, i.e. if
, where
being real and
.
Furthermore, r = 1, ;
,
, (Equation1.11
(1.11)
(1.11) ) relates to the q-analogue of Meijer’s G-function according to Saxena et al. (Citation1983), precisely
where
A full description of Meijer’s G-function, Fox’s H-function, and numerous functions accessible in terms of Fox’s H-function can be seen in research monographs by Mathai & Saxena (Citation1973, Citation1978). Furthermore, the articles of Yadav & Purohit (Citation2006) and Yadav et al. (Citation2008) also provide the basic functions of one variable defined in terms of the functions .
2. The q-generating relations
In this part, using the Iq-function and a family of q-polynomials, we shall derive several bilinear q-generating relations.
Theorem 1.
Let to be positive integers;
,
λ > 0, and let γ to be any arbitrary positive integer. If
is an arbitrary bounded sequence, then the subsequent bilinear q-generating relation follows:
where µ and κ are arbitrary numbers, and
.
Proof.
Taking the L.H.S. of (Equation2.1(2.1)
(2.1) ) written by
and applying the contour integral expression (Equation1.11
(1.11)
(1.11) ) for the q-polynomials
and the Iq-function, we get
Reversing the sequence of summations and integration yields
where (Equation1.12(1.12)
(1.12) ) provides
. Utilizing the q-gamma function relationship, specifically
We find
Utilizing the series reordering a connection once more and altering the order of summations (Srivastava & Manocha, Citation1984)
we obtain
Employing the q-binomial theorem (see Gasper & Rahman, Citation2004) to sum the inner series, specifically
we find that
Currently, by reversing the contour integral and summation’s order and employing the q-identities (see Gasper & Rahman, Citation2004), i.e.
and
we obtain
By analyzing the contour integral of (Equation2.10(2.10)
(2.10) ) in accordance with its description (Equation1.11
(1.11)
(1.11) ) and the notation (Equation2.3
(2.3)
(2.3) ), the intended outcome is obtained. Theorem 1 proof is now complete.
The family of q-polynomials has one value if we determine the bounded sequence along with µ = 0.
so we arrive at the subsequent theorem in light of the right side of (Equation2.1(2.1)
(2.1) ) for δ = 0.
Theorem 2.
Let λ > 0, κ be an arbitrary number and let be satisfied by positive integers
. The q-generating connection for the Iq-function is thus given by
where and
.
3. An integral associated with Iq-function
Following theorems and corollaries, integral is shown involving the Iq-function and family of q-polynomials:
Theorem 3.
If ,
,
, τ be an arbitrary number; assume
to be positive integers;
,
and let γ to be any arbitrary positive integer. If
is an arbitrary bounded sequence, we have the following integral
Proof.
In view of (Equation1.11(1.11)
(1.11) ) and on interchanging the order of integration which is permitted in view of the aforesaid conditions, considering the left-hand side (Equation3.1
(3.1)
(3.1) ) by
, transforms in to the integral
Due to Hann (Citation1949), the integral representation as
Utilizing the above integral in (Equation3.2(3.2)
(3.2) ), we get
In accordance with its description (Equation1.11(1.11)
(1.11) ), the intended outcome is obtained. Proof of the Theorem 3 is now complete.
Theorem 4.
If ,
,
,
,
, τ an arbitrary number and assume
to be positive integers;
,
and let γ to be any arbitrary positive integer. If
is an arbitrary bounded sequence, we have the following integral
Proof.
In view of (Equation1.11(1.11)
(1.11) ) and on interchanging the order of integration which is justified in view of the aforesaid conditions, considering the left-hand side (Equation3.3
(3.3)
(3.3) ) by
, transforms in to the integral
Due to Hann (Citation1949), the integral representation as
Utilizing the above integral formula in (Equation3.4(3.4)
(3.4) ), we obtain
In accordance with its description (Equation1.11(1.11)
(1.11) ), the intended outcome is obtained. Proof of the Theorem 4 is now complete.
Theorem 5.
If ,
, τ an arbitrary number. Suppose
to be positive integers;
,
and γ to be any arbitrary positive integer. If
is an arbitrary bounded sequence, we have the following integral
Proof.
In view of (Equation1.11(1.11)
(1.11) ) and on interchanging the order of integration that is allowed in view of the aforesaid conditions, considering the left-hand side (Equation3.5
(3.5)
(3.5) ) by
, transforms in to the integral
Due to Hann (Citation1949), the integral representation as
Utilizing the above integral in (Equation3.6(3.6)
(3.6) ), making accordance with its description (Equation1.11
(1.11)
(1.11) ), the intended outcome is obtained. Theorem 5 proof is now complete.
Remark: If we set in the Theorems 3-5, the bounded sequence along with
, then for the q-polynomials one has
, and we have the findings stated by Saxena & Kumar (Citation1995), (Equation (3.1)-(3.3), pp. 267).
4. Concluding observations
In this part, we look at a few implications of findings from the preceding section.
If we set r = 1, ; and take (Equation1.14
(1.14)
(1.14) ) into consideration, then Theorem 1 to 5 generate Corollaries 1 to 5, accordingly.
Corollary 1.
Let to be positive integers;
,
λ > 0, and assume γ to be any arbitrary positive integer. If
is an arbitrary bounded sequence, then the subsequent bilinear q-generating relation follows:
where µ and κ are arbitrary numbers, .
Corollary 2.
Let λ > 0, κ an arbitrary number; be satisfied by positive integers
. The q-generating connection for the Iq-function is thus given by
where and
.
Corollary 3.
If ,
,
and assume
to be positive integers;
,
and suppose γ to be any arbitrary positive integer. If
is an arbitrary bounded sequence, we have the following integral
Corollary 4.
If ,
,
,
,
; assume
to be positive integers;
,
and suppose γ to be any arbitrary positive integer. If
is an arbitrary bounded sequence, we have the proceeding integral
Corollary 5.
If ,
and assume
to be positive integers;
,
let γ to be any arbitrary positive integer. If
is an arbitrary bounded sequence, we have the subsequent integral
Remark: If we set in the Corollary 3-5, the bounded sequence along with
, then for the q-polynomials one has
, and we have the findings stated by Saxena et al. (Citation1983) (pp. 140, Equation (3.1)-(3.3)).
If we set r = 1, ;
,
,
, and take (Equation1.16
(1.16)
(1.16) ) into considered, and now, Theorem 1, 2 produce Corollaries 6, 7, correspondingly.
Corollary 6.
Assume to be positive integers;
,
. If
is an arbitrary bounded sequence, then the subsequent bilinear q-generating relation follows:
where κ is an arbitrary number, and
.
Corollary 7.
Suppose κ an arbitrary number and to be positive integers such that
,
then the subsequent bilinear q-generating relation follows:
where .
Again, if we set r = 1, ; ϱ = 1 and take (Equation1.16
(1.16)
(1.16) ) into considered, then Theorem 3 to 5 gives Corollaries 8 to 10, correspondingly.
Corollary 8.
If ,
and assume
to be positive integers;
,
in addition, γ to be any arbitrary positive integer. If
is an arbitrary bounded sequence, we have the following integral
Corollary 9.
If ,
,
,
, assume
to be positive integers;
,
and γ to be any arbitrary positive integer. If
is an arbitrary bounded sequence, we have the proceeding integral
Corollary 10.
If ,
, assume
to be positive integers;
,
γ to be any arbitrary positive integer. If
is an arbitrary bounded sequence, we have the subsequent integral
Furthermore, it is worth noting that in light of subsequent limiting cases:
where
the q-generating relation (Equation2.1(2.1)
(2.1) ) of Theorem 1 provides the q-extension of the present finding acknowledged to Raina (Citation1976) (Equation (2.1), p. 301).
5. Conclusion and future works
Our main conclusion (Theorem 1) can be used to deduce expected bilinear q-generating relations for the product of orthogonal q-polynomials and the Iq-function by giving appropriate special values to the sequence . We use the illustration below to demonstrate this.
(i) Setting γ = 1 and
we discover from (Equation1.10(1.10)
(1.10) ) that
where the q-Laguerre polynomial Srivastava & Agarwal (Citation1989) is denoted by
Theorem 1 thus produces the q-generating connection between the Iq-function and the q-Laguerre polynomial as follows in light of the aforementioned relations:
(ii) Putting γ = 1 and
we get from (Equation1.10(1.10)
(1.10) ) that
where the q-Jacobi polynomial (Mathai & Saxena; Citation1978) is represented by the symbol as
Theorem 1 then gives the q-generating connection between the Iq-function and q-Jacobi polynomial, precisely
The research monograph by Koekoek et al. (Citation2010) and Srivastava & Agarwal (Citation1989) provides a thorough description of several hypergeometric orthogonal q-polynomials. The mathematical descriptions of the q-Jacobi polynomials and q-Laguerre provided by the EquationEquations (5.5)(5.5)
(5.5) and (Equation5.2
(5.2)
(5.2) ), respectively, differ slightly from those provided in the seminal work (Koekoek et al., Citation2010). Consequently, by taking into account the definitions of the q-polynomials provided in Koekoek et al. (Citation2010), one might arrive to results of a similar nature.
We conclude by noting that the general nature of the q-generating relation (Equation2.1(2.1)
(2.1) ) will lead to a number of generating relationships for the product of the q-analogue of the Fox’s H-functions and the orthogonal q-polynomials by appropriately assigning values to the sequence
.
Disclosure statement
No potential conflict of interest was reported by the author(s).
References
- Ahuja, O. P., & Çetinkaya, A. (2021). Faber polynomial expansion for a new subclass of Bi-univalent functions endowed with (p;q) calculus operators. Fundamental Journal of Mathematics and Applications, 4(1), 17–24. https://doi.org/10.33401/fujma.831447
- Al-Omari, S., Suthar, D. L., & Araci, S. (2021). A fractional q-integral operator associated with a certain class of q-Bessel functions and q-generating series. Advances in Difference Equations, 2021(1), Paper No. 441, 13. https://doi.org/10.1186/s13662-021-03594-4
- Araci, S. (2021). Construction of degenerate q-daehee polynomials with weight α and its applications. Fundamental Journal of Mathematics and Applications, 4(1), 25–32. https://doi.org/10.33401/fujma.837479
- Ernst, T. (2003). A method for q-calculus. Journal of Nonlinear Mathematical Physics, 10(4), 487–525. https://doi.org/10.2991/jnmp.2003.10.4.5
- Gasper, G., & Rahman, M. (2004). Basic hypergeometric series (2nd ed.). Cambridge University Press.
- Hann, W. (1949). Beiträge zur Theorie der Heineschen Reihen. Die 24 Integrale der hypergeometrische q-Differenzengleichung. Das q-Analogon der Laplace-Transformation. Mathematische Nachrichten, 2(6), 340–379. https://doi.org/10.1002/mana.19490020604
- Koekoek, R., Lesky, P. A., & Swarttouw, R. F. (2010). Hypergeometric orthogonal polynomials and their q-analogues. Springer Monographs in Mathematics.
- Mathai, A. M., & Saxena, R. K. (1973). Generalized hypergeometric functions with applications in statistics and physical sciences. In Lecture series in mathematics (Vol. 348). Springer-Verlag.
- Mathai, A. M., & Saxena, R. K. (1978). The H-Function with applications in statistics and other Disciplines, Halstedpress [John Wiley and Sons. New York-London-Sidney.
- Meena, S., Bhatter, S., Jangid, K., Purohit, S. D., & Nisar, K. S. (2022). Certain generating functions involving the incomplete I-functions. TWMS Journal of Applied and Engineering Mathematics, 12(3), 985–995.
- Purohit, S. D., Vyas, V. K., & Yadav, R. K. (2012). Bilinear generating relations for a family of q-polynomials and generalised basic hypergeometric function. Acta et Commentationes Universitatis Tartuensis de Mathematica, 16(2),1–9. https://doi.org/10.12697/ACUTM.2012.16.11
- Raina, R. K. (1976). A formal extension of certain generating functions. Proceedings of the National Academy of Sciences, India Section A, 46(IV), 300–304.
- Saxena, R. K., & Kumar, R. (1995). A basic analogue of the generalized H-function. Le Matematiche, 263–271.
- Saxena, R. K., Modi, G. C., & Kalla, S. L. (1983). A basic analogue of Fox’s H-Function. Revista Tecnica de la Facultad de Ingenieria Universidad del Zulia, 6, 139–143.
- Srivastava, H. M., & Agarwal, A. K. (1989). Generating functions for a class of q-polynomials. Annali di Matematica Pura ed Applicata, 154(1), 99–109. https://doi.org/10.1007/BF01790345
- Srivastava, H. M., & Manocha, H. L. (1984). A treatise on generating functions. Halsted Press [John Wiley and Sons].
- Srivastava, H. M., Manocha, H. L., & Kaanoğlu, C. (2010). Some families of generating functions for a certain class of three-variable polynomials. Integral Transforms and Special Functions, 21(12), 885–896. https://doi.org/10.1080/10652469.2010.481439
- Vyas, V. K., Al-Jarrah, A. A., Suthar, D. L. and Abeye, N., Zada, A. (2021). Fractional q-integral operators for the product of a q-polynomial and q-analogue of the I-functions and their applications. Mathematical Problems in Engineering, 2021(1), 1–9. https://doi.org/10.1155/2021/7858331
- Yadav, R. K., & Purohit, S. D. (2006). On applications of Weyl fractional q-integral operator to generalized basic hypergeometric functions. Kyungpook Mathematical Journal, 46, 235–245.
- Yadav, R. K., Purohit, S. D., & Kalla, S. L. (2008). On generalized Weyl fractional q-integral operator involving generalized basic hypergeometric functions. Fractional Calculus and Applied Analysis, 11(2), 129–142.