Generating functions of \((p, q)\)-analogue of aleph-function satisfying Truesdell’s ascending and descending \(F_{p, q}\)-equation. (English) Zbl 1538.33024
Summary: In this paper we have obtained various forms of \((p, q)\)-analogue of Aleph-Function satisfying Truesdell’s ascending and descending \(F_{p, q}\)-equation. These structures have been employed to arrive at certain generating functions for \((p, q)\)-analogue of Aleph-Function. Some specific instances of these outcomes as far as \((p, q)\)-analogue of I-function, H-function and G-functions have likewise been obtained.
MSC:
| 33D05 | \(q\)-gamma functions, \(q\)-beta functions and integrals |
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
Keywords:
\(F_{p, q}\)-equation; generating function; \((p, q)\)-analogue of aleph-function; \((p, q)\)-analogue of I-function; \((p, q)\)-analogue of H-functionReferences:
| [1] | A. Ahmad, R. Jain, D.K. Jain, q-analogue of Aleph-Function and and Its Transformation Formulae with q-Derivative, Journal of Applied Statistics and Probability 6 (2017), 567-575. · doi:10.18576/jsap/060312 |
| [2] | R.K. Saxena, R. Kumar, A basic analogue of the Generalized H-function, Le Mathematic 50 (1995), 263-271. · Zbl 0899.33012 |
| [3] | B.K. Dutta, L.K. Arora, On a Basic Analogue of Generalized H-function, Int. J. Math. Engg. Sci. 1 (2011), 21-30. |
| [4] | R.K. Saxena, G.C. Modi, S.L. Kalla, A basic analogue of Fox’s H-function, Rev. Technology Univ. Zulin 6 (1983), 139-143. · Zbl 0562.33004 |
| [5] | B.M. Agrawal, A unified theory of special functions, A Ph.D. thesis, Vikram University, Ujjain, 1966. |
| [6] | B.M. Agrawal, Rodrigue’s and Schlafli’s formulae for the solution of F- equations, J. Ind. Math. Soc. 41 (1968), 173-177. · Zbl 0183.32401 |
| [7] | V. Kac, P. Cheung, Quantum calculus, Springer, New York, 2002. · Zbl 0986.05001 |
| [8] | I.M. Burban, A.U. Klimyk, (p, q)-differentiation, (p, q)-integration, and (p, q)- hypergeometric functions related to quantum groups, Integral Transforms and Special Functions 2 (1994), 15-36. · Zbl 0823.33010 · doi:10.1080/10652469408819035 |
| [9] | O. Ogievetsky, W.B. Schmidke, J. Wess, B. Zumino, q-Deformed Poincar Algebra, Commun. Math. Phys. 150 (1992), 495. · Zbl 0849.17011 |
| [10] | U. Duran, M. Acikgoz, S. Araci, On (p, q)-Bernoulli,(p, q)-Euler and (p, q)-Genocchi Polynomials, Journal of Computational and Theoretical Nanoscience 13 (2016), 7833-7846. · doi:10.1166/jctn.2016.5785 |
| [11] | T. Kim, J. Choi, Y.H. Kim, C.S. Ryoo, On q-Bernstein and q-Hermite polynomials, Proceedings of the Jangjeon Mathematical Society 14 (2011), 215-221. · Zbl 1256.05024 |
| [12] | P.N. Sadjang, On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas, arXiv preprint arXiv:1309.3934, 2013. · Zbl 1387.30080 |
| [13] | T. Acar, (p, q)-Generalization of Szasz-Mirakyan operators, Math. Methods Appl. Sci. 39 (2016), 2685-2695. · Zbl 1342.41019 · doi:10.1002/mma.3721 |
| [14] | M. Masjed-Jamei, A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions, J. Math. Anal. Appl. 325 (2007), 753-775. · Zbl 1131.33005 · doi:10.1016/j.jmaa.2006.02.007 |
| [15] | F. Soleyman, P. N. Sadjang, M. M. Jamei, I. Area (p, q)-Sturm-Liouville problems and their orthogonal solutions, Math. Methods Appl. Sci. 41 (2018), 8897-9009. · Zbl 1406.90119 · doi:10.1002/mma.5358 |
| [16] | A. Aral, V. Gupta, R.P. Agarwal, Applications of q Calculus in Operator Theory, Springer, New York, 2013. · Zbl 1273.41001 |
| [17] | U. Duran, M. Acikgoz, A. Esi, S. Araci, A Note on the (p, q)-Hermite Polynomials, Appl. Math. Inf. Sci. 12 (2018), 227-231. · doi:10.18576/amis/120122 |
| [18] | P.N. Sadjang, On (p, q)-analogues of the Sumudu transform, http://arxiv:submit/2592285 [math-ph] 2019. |
| [19] | P.N. Sadjang, On Two (p, q)-Analogues Of The Laplace Transform, arXiv:1309.3934v1 [math QA] 2013. · Zbl 1386.44002 |
| [20] | P.N. Sadjang, On the (p, q)-Gamma and the (p, q)-Beta functions, arXiv:1506.07394v1 [math NT] 2015. · Zbl 1387.30080 |
| [21] | A. Ahmad, R. Jain, D.K. Jain, Certain Results of (p, q)-analogue of I-function with (p, q)-derivative, Jnanabha, Special Issue, 37-44, Proceedings: MSSCID-2017, 20th Annual Conference of VPI, Jaipur, India, 2018. · Zbl 1412.33028 |
| [22] | F.H. Jackson, On q-definite integrals, In Quart. J. Pure Appl. Math. 41 (1910), 193-203. · JFM 41.0317.04 |
| [23] | G. Gasper,N. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990. · Zbl 0695.33001 |
| [24] | A. Jain, A.A. Bhat, R. Jain, D.K. Jain, Certain Results of (p, q)-analogue of Aleph-function with (p, q)-Derivative, Journal of Statistics Applications and Probability 7 (2021), 45-52. |
| [25] | M. Masjed-Jamei, F. Soleyman, I. Area, J.J. Nieto, On (p, q)-classical orthogonal polynomials and their characterization theorems, Advances in Difference Equations 186, (2017), 1-17 · Zbl 1444.33007 |
| [26] | C.S. Ryoo, J.Y. Kang, Some symmetric properties and location conjecture of approximate roots for (p, q)-cosine euler polynomials, Symmetry 13 (2021), 1520. |
| [27] | C.S. Ryoo, Some properties of Poly-Cosine tangent and Poly-Sine tangent polynomials, Journal of Applied Mathematics and Informatics 40 (2022), 371-391. · Zbl 1513.11081 · doi:10.14317/JAMI.2022.371 |
| [28] | C.S. Ryoo, J.Y. Kang, Properties of q-Differential Equations of Higher Order and Visualization of Fractal Using q-Bernoulli Polynomials, Fractal and Fractional 6 (2022), 296. |
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