$\pi$-Formulae from dual series of the Dougall theorem

  • W. Chu School Math. and Statistics, Zhoukou Normal Univ., Henan, China and Univ. Salento, Italy
Keywords: Classical hypergeometric series; The Dougall summation theorem; Gould-Hsu inverse series relations; Ramanujan's series for $1/\pi$; Guillera's series for $1/\pi^2$.


UDC 517.5

By means of the extended Gould–Hsu inverse series relations, we find that the dual relation of Dougall's summation theorem for the well-poised $_7F_6$-series can be used to construct numerous interesting Ramanujan-like infinite-series expressions  for $\pi^{\pm1}$ and $\pi^{\pm2},$ including an elegant formula  for $\pi^{-2}$ due to Guillera.


V. Adamchik, S. Wagon, $pi$: A 2000-year search changes direction, Math. Educ. and Res., 5, № 1, 11–19 (1996).

D. Bailey et al., On the rapid computation of various polylogarithmic constants, Math. Comp., 66, № 218, 903–913 (1997). DOI: https://doi.org/10.1090/S0025-5718-97-00856-9

W. N. Bailey, Generalized hypergeometric series, Cambridge Univ. Press, Cambridge (1935).

D.~M.~Bressoud, A matrix inverse, Proc. Amer. Math. Soc., 88, № 3, 446–448 (1983). DOI: https://doi.org/10.1090/S0002-9939-1983-0699411-9

Yu.~A.~Brychkov, Handbook of special functions: derivatives, integrals, series and other formulas, CRC Press, Boca Raton, FL (2008). DOI: https://doi.org/10.1201/9781584889571

L. Carlitz, Some inverse series relations, Duke Math. J., 40, 893–901 (1973). DOI: https://doi.org/10.1215/S0012-7094-73-04083-0

X.~Chen, W.~Chu, Closed formulae for a class of terminating $_3F_2(4)$-series, Integral Transforms and Spec. Funct., 28, № 11, 825–837 (2017). DOI: https://doi.org/10.1080/10652469.2017.1376194

X.~Chen, W.~Chu, Terminating $_3F_2(4)$-series extended with three integer parameters, J. Difference Equat. and Appl., 24, № 8, 1346–1367 (2018). DOI: https://doi.org/10.1080/10236198.2018.1485668

W.~Chu, Inversion techniques and combinatorial identities, Boll. Unione Mat. Ital., Ser. 7, 737–760 (1993).

W.~Chu, Inversion techniques and combinatorial identities: strange evaluations of hypergeometric series, Pure Math. and Appl., 4, № 4, 409–428 (1993).

W. Chu, A new proof for a terminating ``strange'' hypergeometric evaluation of Gasper and Rahman conjectured by Gosper, C. R. Math. Acad. Sci. Paris, 318, 505–508 (1994).

W.~Chu, Inversion techniques and combinatorial identities: a quick introduction to the hypergeometric evaluations, Math. Appl., 283, 31–57 (1994). DOI: https://doi.org/10.1007/978-1-4613-3635-8_3

W.~Chu, Inversion techniques and combinatorial identities: basic hypergeometric identities, Publ. Math. Debrecen, 44, № 3/4, 301–320 (1994). DOI: https://doi.org/10.5486/PMD.1994.1367

W.~Chu, Inversion techniques and combinatorial identities: strange evaluations of basic hypergeometric series, Compos. Math., 91, 121–144 (1994).

W.~Chu, Inversion techniques and combinatorial identities: a unified treatment for the $_7F_6$-series identities, Collect. Math., 45, № 1, 13–43 (1994).

W.~Chu, Inversion techniques and combinatorial identities:

Jackson's $q$-analogue of the Dougall–Dixon theorem and the dual formulae, Compos. Math., 95, 43–68 (1995).

W.~Chu, Duplicate inverse series relations and hypergeometric evaluations with $z=1/4$, Boll. Unione Mat. Ital., Ser. 8, 585–604 (2002).

W.~Chu, Inversion techniques and combinatorial identities: balanced hypergeometric series, Rocky Mountain J. Math., 32, № 2, 561–587 (2002). DOI: https://doi.org/10.1216/rmjm/1030539687

W.~Chu, $q$-Derivative operators and basic hypergeometric series, Results Math., 49, № 1-2, 25–44 (2006). DOI: https://doi.org/10.1007/s00025-006-0209-1

W.~Chu, X.~Wang, Summation formulae on Fox–Wright $Psi$-functions, Integral Transforms and Spec. Funct., 19, № 8, 545–561 (2008). DOI: https://doi.org/10.1080/10652460802091575

W.~Chu, W. Zhang, Accelerating Dougall's $_5F_4$-sum and infinite series involving $pi$, Math. Comp., 83, № 285, 475–512 (2014). DOI: https://doi.org/10.1090/S0025-5718-2013-02701-9

J. Dougall, On Vandermonde's theorem, and some more general expansions, Proc. Edinb. Math. Soc., 25, 114–132 (1907). DOI: https://doi.org/10.1017/S0013091500033642

I. Gessel, Finding identities with the WZ method, J. Symbolic Comput., 20, № 5-6, 537–566 (1995). DOI: https://doi.org/10.1006/jsco.1995.1064

I. Gessel, D. Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal., 13, № 2, 295–308 (1982). DOI: https://doi.org/10.1137/0513021

I.~Gessel, D.~Stanton, Applications of $q$-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc., 277, № 1, 173–201 (1983). DOI: https://doi.org/10.1090/S0002-9947-1983-0690047-7

I. Gessel, D. Stanton, Another family of $q$-Lagrange inversion formulas, Rocky Mountain J. Math., 16, № 2, 373–384 (1986). DOI: https://doi.org/10.1216/RMJ-1986-16-2-373

H.~W.~Gould, L.~C.~Hsu, Some new inverse series relations, Duke Math.~J., 40, 885–891 (1973). DOI: https://doi.org/10.1215/S0012-7094-73-04082-9

J. Guillera, About a new kind of Ramanujan-type series, Exp. Math., 12, № 4, 507–510 (2003). DOI: https://doi.org/10.1080/10586458.2003.10504518

J. Guillera, Generators of some Ramanujan formulas, Ramanujan J., 11, № 1, 41–48 (2006). DOI: https://doi.org/10.1007/s11139-006-5306-y

J. Guillera, Hypergeometric identities for 10 extended Ramanujan-type series, Ramanujan J., 15, № 2, 219–234 (2008). DOI: https://doi.org/10.1007/s11139-007-9074-0

E.~D.~Rainville, Special functions, The Macmillan Co., New York (1960).

S. Ramanujan, Modular equations and approximations to $pi$, Quart. J. Math., 45, 350–372 (1914).

K. R. Stromberg, An introduction to classical real analysis, Wadsworth, INC, Belmont, California (1981).

How to Cite
Chu, W. “$\pi$-Formulae from Dual Series of the Dougall Theorem”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 12, Jan. 2023, pp. 1686 -08, doi:10.37863/umzh.v74i12.6587.
Research articles