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Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation



Altaf Ahmad Bhat, Renu Jain and Deepak Kumar Jain
 
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ABSTRACT

Various forms of q-analogue of I-function satisfying Truesdell’s descending Fq-equation have been studied and various generating functions by the application of these various forms have been derived. The aim of this study was to produce q-analogue of I-function which fulfilling Truesdell’s Ascending Fq-equation. In this study, various forms of q-analogue of I-function satisfying Truesdell’s ascending Fq-equation have been obtained. Certain generating functions for q-analogue of I-function have been derived by using these forms. Further, some particular cases of these results in terms of q-analogue of H-and G-functions which appear to be new have also been obtained.

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  How to cite this article:

Altaf Ahmad Bhat, Renu Jain and Deepak Kumar Jain, 2018. Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation. Asian Journal of Mathematics & Statistics, 11: 40-45.

DOI: 10.3923/ajms.2018.40.45

URL: https://scialert.net/abstract/?doi=ajms.2018.40.45
 
Received: September 30, 2018; Accepted: November 06, 2018; Published: January 05, 2019


Copyright: © 2018. This is an open access article distributed under the terms of the creative commons attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

INTRODUCTION

The concept of basic hypergeometric functions has been studied by many authors and some new generalized forms of these functions have been derived. So, it is significant to study these functions due to their applications in the fields like engineering and physical sciences1.

Some basic hypergeometric functions are Meijer’s G-function, Fox’s H-function, Mac-Roberts’s E-function, Saxena’s I-function and their q-analogues. The q-analogue of I-function have been introduced in terms of Mellin-Barnes type basic contour integral by Saxena et al.2 as:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(1)

where, αj, βj, αji βji, are real and positive, aj, bj, a, bji are complex numbers and:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation

where, L is representing a contour of integration ranging from -i∞ to i∞ in such a way so that all poles of Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation1<j<m are to right and those of Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation1<j<n. The integral converges if Re [s log (x)-log sin πs}<0, for large values of |s| on the contour L.

Setting r = 1, Ai = A, Bi = B in Eq. 1 we get q-analogue of H-function defined by Saxena et al.2 as follows:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(2)

Additional if we put αj = βj = 1, Eq. 2 reduces to the basic analogue of Meijer’s G-function specified by Saxena et al.2:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(3)

Basic analogue of I-function in terms of Gamma function is defined by Ahmad et al.3 as follows:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(4)

Certain expansion formulae for a basic analogue of I-function defined by Ahmad et al.3 in terms of basic analogue of Gamma function have been derived by Ahmad et al.4.

To achieve the unification of special functions, Truesdell5 has introduced the theory which provided a number of results for special functions satisfying the so called Truesdell’s F-equation. Agrawal6 extended the concept further and derived results for descending F-equation. Various properties like orthogonality, Rodrigue’s and Schalafli’s formulae for F-equation, which turn out to be special functions have been obtained by Agrawal6.

The function F(z, α) is supposed to satisfy the ascending F-equation if:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(5)

Truesdell5 has obtained following generating functions using Taylor’s series for F(z, a) satisfying ascending F-equation:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(6)

The q-derivative of Eq. 5 can be written as:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(7)

Ahmad et al.7 obtained generating functions of q-analogue of I-function satisfying Truesdell’s descending Fq-equation. Jain et al.8 derived some generating functions of q-analogue of Mittag-Leffler function and Hermite polynomial satisfying Truesdell’s ascending and descending Fq-equation. In this connection, it have obtained various forms of I-function which satisfies Truesdell's Fq-equation and have obtained various generating functions by employing these forms.

Following results have been used of multiplication formulae for q-analogue of Gamma functions to obtain main the consequences of this study:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(8)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(9)

In the results that follow, by Δ(μ, α) we shall mean the array of μ parameters:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(10)

and:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(11)

The main objective of this study was to generate functions of q-analogue of I-function that satisfying Truesdell’s ascending Fq-equation. In this study, it have extended Truesdell’s F-equation to its q-analogue and named the corresponding equation as Fq-equation. It have further derived various forms of q-analogue of I-function satisfying Truesdell’s ascending Fq-equation. These forms have been employed to arrive at certain generating functions for q-analogue of I-function. Some particular cases of these results which appear to be new have also been obtained.

GENERATING FUNCTIONS FOR Q-ANALOGUE OF I-FUNCTION

Various forms of I-function satisfying Truesdell's ascending F-equation have been obtained by Jain et al.9 and using this, different forms of q-analogue of I-function which satisfy Truesdell's ascending Fq-equation have been established:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(12)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(13)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(14)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(15)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(16)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(17)

Suppose that the form Eq. 12 is A (z, α), then replace q-analogue of I-function by the definition (Eq. 4) and interchanging order of integration and differentiation, which is justified under the convergence conditions (Eq. 3), it observed that:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(18)

Now results Eq. 8 and 9 lead to two very important identities:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(19)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(20)

Using these identities Eq. 19 and 20 we see that Eq. 18 takes the form:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(21)

This is the Truesdell’s form of ascending Fq-equation.

In the same way forms Eq. 13-17 can be exposed to satisfy Truesdell’s ascending Fq-equation.

In this part it utilized forms Eq. 12-17, to found the following generating functions for q-analogue of I-functions using Truesdell’s ascending Fq-equation technique:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(22)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(23)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(24)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(25)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(26)

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(27)

Proof: To found Eq. 22 it substitute the form Eq. 12 in Truesdell’s ascending F-equation (Eq. 6) and replace by Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equationby x in sequence to obtain the necessary consequence. Likewise, consequence Eq. 26 can be proved by substituting the form Eq. 16 in Truesdell’s ascending F-equation (Eq. 6) and with same substitution.

To found Eq. 23 it substitute the form Eq. 13 in Truesdell’s ascending F-equation (Eq. 6) and substitute by Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equationby x in series to get the mandatory result. Similarly, result (Eq. 24) can be proved by substituting the form Eq. 14 in Truesdell’s ascending F-equation (Eq. 6) and by same substitution.

To establish Eq. 25 we substitute the form Eq. 15 in Truesdell’s ascending F-equation (Eq. 6) and alternate by Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation by x in series to get the necessary outcome. Correspondingly, result (Eq. 27) can be established by substituting the form Eq. 17 in Truesdell’s ascending F-equation (Eq. 6) and via same substitution.

Special cases: These consequences capitulate as special cases of certain generating function for q-analogue of Fox’s H-function2 and q-analogue of Meijer’s G-function2.

If we supposed then the sequence (Eq. 22) diminishes to generating function of q-analogue of Fox’s H-function:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(28)

Again taking in Eq. 28, it gives Meijer’s G-function as:

Image for - Generating q-Analogue of I-Function Satisfying Truesdell's Ascending Fq-Equation
(29)

In the same way, Eq. 23-27 can be used to acquiesce apparently new and interesting consequences for q-analogue of Fox’s H-function and Meijer’s G-function2.

CONCLUSION

This study presented various forms of q-analogue of I-function satisfying Truesdell’s ascending Fq-equation. These forms have been employed to arrive at certain generating functions for q-analogue of basic analogue of I-function.

The results proved in this study along with their particular cases are believed to be new. As these functions have well established as applicable functions these results are likely to contribute significantly in certain application of the theory of q-calculus.

SIGNIFICANCE STATEMENT

This study discovers various forms of q-analogue of I-function satisfying Truesdell’s ascending Fq-equation. These forms have been employed to arrive at certain generating functions for q-analogue of basic analogue of I-function. As these functions have well established as applicable functions these results are likely to contribute significantly in certain application of the theory of q-calculus.

ACKNOWLEDGMENT

The authors are immensely grateful to the worthy referees for some useful and valuable suggestions for the improvement of this paper which led to a better presentation.

REFERENCES

1:  Ganie, H.A. and B.A. Chat, 2018. Skew Laplacian energy of digraphs. Afrika Matematika, 29: 499-507.
CrossRef  |  Direct Link  |  

2:  Saxena, R.K., G.C. Modi and S.L. Kalla, 1983. A basic analogue of Fox's H-function. Rev. Tec. Ing. Univ., Zulin, 6: 139-143.
Direct Link  |  

3:  Ahmad, F., R. Jain and D.K. Jain, 2012. Q-integral and Basic Analogue of I-function. Asian J. Mathe. Stat., 5: 99-103.
CrossRef  |  Direct Link  |  

4:  Ahmad, A., R. Jain and D.K. Jain, 2017. Certain expansion formulae involving a basic analogue of I-function. J. Indian Acad. Math. Indore, 39: 129-136.

5:  Truesdell, C., 1948. An Essay Toward the Unified Theory of Special Functions. Princeton University Press, USA

6:  Agrawal, B.M., 1966. A unified theory of special functions. Ph.D. Thesis, Vikram University, Ujjain.

7:  Ahmad, A., R. Jain and D.K. Jain, 2017. Generating functions of q-analogue of I-function satisfying Truesdell's descending Fq-equation. J. New Theory, 19: 48-55.
Direct Link  |  

8:  Jain, R., A. Ahmad and D.K. Jain, 2017. Some generating functions of q-analogue of mittag-leffler function and hermite polynomial satisfying truesdell’s ascending and descending Fq-equation. GAMS J. Math. Math. Biosci., Vol. 6.

9:  Jain, R., P.K. Chaturvedi and D.K. Jain, 2008. Some generating functions of I-functions involving Truesdell's F-equation technique. Rajasthan Ganit Parishad, Vol. 21, No. 2.

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