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Expansions over Legendre polynomials and infinite double series identities

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Abstract

An expansion formula of the classical Gaussian \(_2F_1\)-series in terms of Legendre polynomials due to Holdeman (J Math Phys 11:114–117, 1970) is extended to generalized hypergeometric series of higher order. Its special cases together with Holdeman’s formula are employed to derive explicit Fourier–Legendre series for forty \(_2F_1\)-functions. By applying a “product integration" method to these \(_2F_1\)-expressions, we establish numerous identities for double series involving multinomial coefficients, including a remarkable formula for \(\zeta (3)/\pi ^2\) found recently by the second author.

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References

  1. Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 54, 319 (1985)

    MATH  Google Scholar 

  2. Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935)

    MATH  Google Scholar 

  3. Campbell, J.M.: Ramanujan-like series for \(\frac{1}{\pi }\) involving harmonic numbers. Ramanujan J. 46, 373–387 (2018)

    Article  MATH  Google Scholar 

  4. Campbell, J.M.: New families of double hypergeometric series for constants involving \(\frac{1}{\pi ^2}\). Ann. Polon. Math. 126, 1–20 (2021)

    Article  Google Scholar 

  5. Campbell, J.M., Chu, W.: Double series transforms derived from Fourier-Legendre theory. Commun. Korean Math. Soc. 37, 551–566 (2022)

    MATH  Google Scholar 

  6. Campbell, J.M., D’Aurizio, J., Sondow, J.: On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre expansions. J. Math. Anal. Appl. 479, 90–121 (2019)

    Article  MATH  Google Scholar 

  7. Chu, W.: Analytical formulae for extended \(_3F_2\)-series of Watson-Whipple-Dixon with two extra interger parameters. Math. Comput. 81(277), 467–479 (2012)

    Article  MATH  Google Scholar 

  8. Dong, S.-H., Lemus, R.: The overlap integral of three associated Legendre polynomials. Appl. Math. Lett. 15, 541–546 (2002)

    Article  MATH  Google Scholar 

  9. González, M.O.: Elliptic integrals in terms of Legendre polynomials. Proc. Glasgow Math. Assoc. 2, 97–99 (1954)

    Article  MATH  Google Scholar 

  10. Holdeman, J.T.: Legendre polynomial expansions of hypergeometric functions with applications. J. Math. Phys. 11, 114–117 (1970)

    Article  MATH  Google Scholar 

  11. Karlsson, P.W.: Some reducible generalized Kampé de Fériet functions. J. Math. Anal. Appl. 96, 546–550 (1983)

    Article  MATH  Google Scholar 

  12. Karlsson, P.W.: Two hypergeometric summation formulae related to \(9-j\) coefficients. J. Phys. A 27(27), 6943–6945 (1994)

    Article  MATH  Google Scholar 

  13. Karlsson, P.W.: Some formulae for double Clausenian functions. J. Comput. Appl. Math. 118(1/2), 203–213 (2000)

    Article  MATH  Google Scholar 

  14. Koekoek, R., Swarttouw, R. F.: The Askey-scheme of hypergeometric orthogonal polynomials and its \(q\)-analogue Report 98-17 (1998), Delft University of Technology

  15. Lievensa, S., Van der Jeugt, J.: Transformation formulas for double hypergeometric series related to \(9-j\) coefficients and their basic analogs. J. Math. Phys. 42(11), 5417–5430 (2001)

    Article  MATH  Google Scholar 

  16. Pitre, S.N., Van der Jeugt, J.: Transformation and summation formulas for Kampé de Fériet Series \(F_{1:1}^{0:3}(1,1)\). J. Math. Anal. Appl. 202, 121–132 (1996)

    Article  MATH  Google Scholar 

  17. Qureshi, M.I., Khan, M.K.: Some quadratic transformations and reduction formulas associated with hypergeometric functions. Appl. Appl. Math. 6, 71–86 (2020)

    MATH  Google Scholar 

  18. Rainville, E.D.: Special Functions. The Macmillan Company, New York (1960)

    MATH  Google Scholar 

  19. Singal, R.P.: A transformation formula for double hypergeometric series Rocky Mountain. J. Math. 3, 377–381 (1973)

    MATH  Google Scholar 

  20. Van der Jeugt, J., Pitre, S.N., Rao, K.S.: Transformation and summation formulas for double hypergeometric series. J. Comput. Appl. Math. 83, 185–193 (1997)

    Article  MATH  Google Scholar 

  21. Wang, X.Y., Chu, W.: Further Ramanujan-like series containing harmonic numbers and squared binomial coefficients. Ramanujan J. 52(3), 641–668 (2020)

    Article  MATH  Google Scholar 

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Acknowledgements

The authors are sincerely grateful to anonymous reviewers for their careful reading, critical comments, and valuable suggestions that contributed significantly to improving the manuscript during the revision.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, Henan, China

    Wenchang Chu

  2. Department of Mathematics and Physics, University of Salento, P. O. Box 193, 73100, Lecce, Italy

    Wenchang Chu

  3. Department of Mathematics and Statistics, York University, Toronto, Canada

    John M. Campbell

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  1. Wenchang Chu
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  2. John M. Campbell
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Correspondence to John M. Campbell.

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Chu, W., Campbell, J.M. Expansions over Legendre polynomials and infinite double series identities. Ramanujan J 60, 317–353 (2023). https://doi.org/10.1007/s11139-022-00663-4

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