Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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GENERATING FUNCTIONS OF (p, q)-ANALOGUE OF ALEPH-FUNCTION SATISFYING TRUESDELL'S ASCENDING AND DESCENDING Fp,q-EQUATION

  • ALTAF A. BHAT (Department of Mathematical Sciences, Islamic University of Science and Technology) ;
  • M. YOUNUS BHAT (Department of Mathematical Sciences, Islamic University of Science and Technology) ;
  • H. MAQBOOL (Department of Mathematical Sciences, Islamic University of Science and Technology) ;
  • D.K. JAIN (Department of Engineering Mathematics and Computing, MITS)
  • Received : 2022.06.11
  • Accepted : 2022.09.27
  • Published : 2023.03.30
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Abstract

In this paper we have obtained various forms of (p, q)-analogue of Aleph-Function satisfying Truesdell's ascending and descending Fp,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.

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Acknowledgement

We are grateful to the anonymous referees for carefully reading the manuscript, detecting many mistakes and for offering valuable comments and suggestions which enabled us to substantially improve the paper. This work is supported by the Research project (JKST&IC/SRE/J/357-60) provided by JKST&IC, UT of J&K, India.

References

  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. https://doi.org/10.18576/jsap/060312
  2. R.K. Saxena, R. Kumar, A basic analogue of the Generalized H-function, Le Mathematic 50 (1995), 263-271.
  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.
  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.
  7. V. Kac, P. Cheung, Quantum calculus, Springer, New York, 2002.
  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. https://doi.org/10.1080/10652469408819035
  9. O. Ogievetsky, W.B. Schmidke, J. Wess, B. Zumino, q-Deformed Poincar Algebra, Commun. Math. Phys. 150 (1992), 495.
  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. https://doi.org/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.
  12. P.N. Sadjang, On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas, arXiv preprint arXiv:1309.3934, 2013.
  13. T. Acar, (p, q)-Generalization of Szasz-Mirakyan operators, Math. Methods Appl. Sci. 39 (2016), 2685-2695. https://doi.org/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. https://doi.org/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. https://doi.org/10.1002/mma.5358
  16. A. Aral, V. Gupta, R.P. Agarwal, Applications of q Calculus in Operator Theory, Springer, New York, 2013.
  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. https://doi.org/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.
  20. P.N. Sadjang, On the (p, q)-Gamma and the (p, q)-Beta functions, arXiv:1506.07394v1 [math NT] 2015.
  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.
  22. F.H. Jackson, On q-definite integrals, In Quart. J. Pure Appl. Math. 41 (1910), 193-203.
  23. G. Gasper,N. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
  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. https://doi.org/10.18576/jsap/100105
  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 https://doi.org/10.1186/s13662-017-1236-9
  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. https://doi.org/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.