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Year 2023, Volume: 52 Issue: 2, 391 - 397, 31.03.2023
https://doi.org/10.15672/hujms.1116891

Abstract

References

  • [1] W.N. Bailey, Some identities involving generalized hypergeometric series, Proc. London Math. Soc. 29, 503–516, 1929.
  • [2] W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935.
  • [3] Yury A. Brychkov, Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas, CRC Press, Taylor & Francis Group, Boca Raton, London, New York, 2008.
  • [4] L. Carlitz, Summation of a special $_4F_3$, Boll. Union Mat. Ital. 18, 90–93, 1963.
  • [5] W. Chu, Inversion techniques and combinatorial identities: A quick introduction to hypergeometric evaluations, Math. Appl. 283, 31–57, 1994.
  • [6] W. Chu, Inversion techniques and combinatorial identities: a unified treatment for the $_7F_6$-series identities, Collect. Math. 45 (1), 13–43, 1994.
  • [7] W. Chu, Binomial convolutions and hypergeometric identities, Rend. Circolo Mat. Palermo (serie 2) 43, 333–360, 1994.
  • [8] W. Chu, Inversion techniques and combinatorial identities: balanced hypergeometric series, Rocky Mountain J. Math. 32 (2), 561–587, 2002.
  • [9] W. Chu, Analytical formulae for extended $_3F_2$-series of Watson–Whipple–Dixon with two extra integer parameters, Math. Comp. 81 (277), 467–479, 2012.
  • [10] W. Chu, Terminating $_4F_3$-series extended with two integer parameters, Integral Transforms Spec. Funct. 27 (10), 794–805, 2016.
  • [11] J. Dougall, On Vandermonde’s theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25, 114–132, 1907.
  • [12] I.M. Gessel, Finding identities with the WZ method, J. Symbolic Comput. 20 (5/6), 537–566, 1995.
  • [13] I.M. Gessel and D. Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (2), 295–308, 1982.
  • [14] A. Ishkhanyan and C. Cesarano, Generalized-hypergeometric solutions of the general Fuchsian linear ODE having five regular singularities, Axioms 8, (102), 2019.
  • [15] A. Lupica, C. Cesarano, F. Crisanti, and A. Ishkhanyan, Analytical solution of the three-dimensional Laplace equation in terms of linear combinations of hypergeometric functions, Mathematics 9, (3316), 2021.
  • [16] I.D. Mishev, Relations for a class of terminating $_4F_3(4)$ hypergeometric series, Integral Transforms Spec. Funct. 33 (3), 199–215, 2022.
  • [17] C.Y. Wang and X.J. Chen, A short proof for Gosper’s $_7F_6$-series conjecture, J. Math. Anal. Appl. 422 (2), 819–824, 2015.
  • [18] F.J.W. Whipple, A group of generalized hypergeometric series: Relations between 120 allied series of type $F[a,b,c;d,e]$, Proc. London Math. Soc. (Ser.2) 23, 104–114, 1925.
  • [19] F.J.W. Whipple, On well–poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc. (Ser.2) 24, 247–263, 1926.
  • [20] D. Zeilberger, Forty “strange" computer–discovered and computer–proved (of course) hypergeometric series evaluations, Available at http://www.math.rutgers.edu/ ~zeilberg/ekhad/ekhad.html.

Transformation formulae for terminating balanced $_4F_3$-series and implications

Year 2023, Volume: 52 Issue: 2, 391 - 397, 31.03.2023
https://doi.org/10.15672/hujms.1116891

Abstract

A new transformation from terminating balanced $_4F_3$-series to $_3F_2$-series is proved that contains a few known summation formulae as special cases. By means of Whipple's transformation, further closed form evaluations are given for terminating well--poised $_7F_6$-series as applications.

References

  • [1] W.N. Bailey, Some identities involving generalized hypergeometric series, Proc. London Math. Soc. 29, 503–516, 1929.
  • [2] W.N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935.
  • [3] Yury A. Brychkov, Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas, CRC Press, Taylor & Francis Group, Boca Raton, London, New York, 2008.
  • [4] L. Carlitz, Summation of a special $_4F_3$, Boll. Union Mat. Ital. 18, 90–93, 1963.
  • [5] W. Chu, Inversion techniques and combinatorial identities: A quick introduction to hypergeometric evaluations, Math. Appl. 283, 31–57, 1994.
  • [6] W. Chu, Inversion techniques and combinatorial identities: a unified treatment for the $_7F_6$-series identities, Collect. Math. 45 (1), 13–43, 1994.
  • [7] W. Chu, Binomial convolutions and hypergeometric identities, Rend. Circolo Mat. Palermo (serie 2) 43, 333–360, 1994.
  • [8] W. Chu, Inversion techniques and combinatorial identities: balanced hypergeometric series, Rocky Mountain J. Math. 32 (2), 561–587, 2002.
  • [9] W. Chu, Analytical formulae for extended $_3F_2$-series of Watson–Whipple–Dixon with two extra integer parameters, Math. Comp. 81 (277), 467–479, 2012.
  • [10] W. Chu, Terminating $_4F_3$-series extended with two integer parameters, Integral Transforms Spec. Funct. 27 (10), 794–805, 2016.
  • [11] J. Dougall, On Vandermonde’s theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25, 114–132, 1907.
  • [12] I.M. Gessel, Finding identities with the WZ method, J. Symbolic Comput. 20 (5/6), 537–566, 1995.
  • [13] I.M. Gessel and D. Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (2), 295–308, 1982.
  • [14] A. Ishkhanyan and C. Cesarano, Generalized-hypergeometric solutions of the general Fuchsian linear ODE having five regular singularities, Axioms 8, (102), 2019.
  • [15] A. Lupica, C. Cesarano, F. Crisanti, and A. Ishkhanyan, Analytical solution of the three-dimensional Laplace equation in terms of linear combinations of hypergeometric functions, Mathematics 9, (3316), 2021.
  • [16] I.D. Mishev, Relations for a class of terminating $_4F_3(4)$ hypergeometric series, Integral Transforms Spec. Funct. 33 (3), 199–215, 2022.
  • [17] C.Y. Wang and X.J. Chen, A short proof for Gosper’s $_7F_6$-series conjecture, J. Math. Anal. Appl. 422 (2), 819–824, 2015.
  • [18] F.J.W. Whipple, A group of generalized hypergeometric series: Relations between 120 allied series of type $F[a,b,c;d,e]$, Proc. London Math. Soc. (Ser.2) 23, 104–114, 1925.
  • [19] F.J.W. Whipple, On well–poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc. (Ser.2) 24, 247–263, 1926.
  • [20] D. Zeilberger, Forty “strange" computer–discovered and computer–proved (of course) hypergeometric series evaluations, Available at http://www.math.rutgers.edu/ ~zeilberg/ekhad/ekhad.html.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Wenchang Chu 0000-0002-8425-212X

Publication Date March 31, 2023
Published in Issue Year 2023 Volume: 52 Issue: 2

Cite

APA Chu, W. (2023). Transformation formulae for terminating balanced $_4F_3$-series and implications. Hacettepe Journal of Mathematics and Statistics, 52(2), 391-397. https://doi.org/10.15672/hujms.1116891
AMA Chu W. Transformation formulae for terminating balanced $_4F_3$-series and implications. Hacettepe Journal of Mathematics and Statistics. March 2023;52(2):391-397. doi:10.15672/hujms.1116891
Chicago Chu, Wenchang. “Transformation Formulae for Terminating Balanced $_4F_3$-Series and Implications”. Hacettepe Journal of Mathematics and Statistics 52, no. 2 (March 2023): 391-97. https://doi.org/10.15672/hujms.1116891.
EndNote Chu W (March 1, 2023) Transformation formulae for terminating balanced $_4F_3$-series and implications. Hacettepe Journal of Mathematics and Statistics 52 2 391–397.
IEEE W. Chu, “Transformation formulae for terminating balanced $_4F_3$-series and implications”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 2, pp. 391–397, 2023, doi: 10.15672/hujms.1116891.
ISNAD Chu, Wenchang. “Transformation Formulae for Terminating Balanced $_4F_3$-Series and Implications”. Hacettepe Journal of Mathematics and Statistics 52/2 (March 2023), 391-397. https://doi.org/10.15672/hujms.1116891.
JAMA Chu W. Transformation formulae for terminating balanced $_4F_3$-series and implications. Hacettepe Journal of Mathematics and Statistics. 2023;52:391–397.
MLA Chu, Wenchang. “Transformation Formulae for Terminating Balanced $_4F_3$-Series and Implications”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 2, 2023, pp. 391-7, doi:10.15672/hujms.1116891.
Vancouver Chu W. Transformation formulae for terminating balanced $_4F_3$-series and implications. Hacettepe Journal of Mathematics and Statistics. 2023;52(2):391-7.