On Some Classes of Series Representations for $1/\pi$ and $\pi^2$

Year 2021, Volume: 9 Issue: 4, 176 - 184, 31.12.2021

Hakan Küçük , Sezer Sorgun

https://doi.org/10.36753/mathenot.795582

Abstract

We propose some classes of series representations for $1/\pi$ and $\pi^2$ by using a new WZ-pair. As examples, among many others, we prove that
\begin{equation*}
\frac{3}{2}\sum_{n=1}^{\infty}\frac{n}{16^n(n+1)(2n-1)}\binom{2n}{n}^2=\frac{1}{\pi},
\end{equation*}
\begin{equation*}
1-\frac{1}{4}\sum_{n=0}^{\infty}\frac{3n+2}{(n+1)^2}\binom{2n}{n}^2 \frac{1}{16^n}=\frac{1}{\pi}
\end{equation*}
and
$$
4\sum_{n=0}^{\infty}\frac{1}{(n+1)(2n+1)}\frac{4^n}{ \binom{2 n}{n}}=\pi^2.
$$
Furthermore, our results lead to new combinatorial identities and binomial sums involving harmonic numbers.

Keywords

Ramanujan-type series, WZ pair, Combinatorial identities, Binomial sums, Ekhad package

References

• [1] G.Bauer,Vondencoefficientenderreihenvonkugelfunctioneneinervariabeln,J.ReineAngew.Math.,56(1859) 101-121.
• [2] N.D. Baruah, B. C. Berndt and H. H. Chan, Ramanujan’s Series for 1/π: A Survey. Amer. Math. Monthly, Vol. 116, No. 7 (Aug. - Sep., 2009), pp. 567-587.
• [3] J.M.BorweinandP.B.Borwein,PiandtheAGM:AstudyinAnalyticNumberTheoryandComputational Complexity, Wiley, New York, 1987.
• [4] J.M.BorweinandP.B.Borwein,ClassnumberthreeRamanujantypeseriesfor1/π,J.Comput.Appl.Math., 46(1993) 281-290.
• [5] J.M.BorweinandP.B.Borwein,MoreRamanujan-typeseriesfor1/π,InRamanujanRevisited,G.E.Andrews, R. A. Askey, B. C. Berndt, K. G Ramanathan and R. A. Rankin (edts), Academic Press, Boston, 1988, 359-374.
• [6] J. M. Borwein and P. B. Borwein, Ramanujan’s rational and algebraic series for 1/π, J. Indian Math. Soc., 51(1987), 147-160.
• [7] , H. H. Chan, S. H. Chan and Z. Liu, Domb’s numbers and Ramanujan-Sato type series for 1/π, Adv. Math., v.186, no.2, 2004, 396-410.
• [8] H.H.Chan,J.WanandW.Zudilin,LegendrepolynomialsandRamanujan-typeseriesfor1/π,IsraelJ.ofMath., v.194, no.1, 2013, 183-207.
• [9] D.V.ChudnovskyandG.V.Chudnovsky,InRmanujanRevisited,ProceedingsofthecentenaryConference (Urbana-Champaign), G. E. Andrews, R. A. Askey, B. C. Berndt, K. G Ramanathan and R. A. Rankin (edts), Academic Press, Boston, 1988, 375-472.
• [10] J.W.L.Glaisher,Onseriesfor1/πand1/π2,Quart.,J.PureAppl.Math.,37(1905)173-198.
• [11] J. Guillera, Dougall’s 5F4 sum and the WZ algorithm, Ramanujan J. v.46, no.3, no.1, 2018, 667-675.
• [12] J. Guillera, Proofs of some Ramanujan series for 1/π using a program due to Zeilberger, J. Difference Equ. Appl., v.24, no.10, 2018, 1643-1648.
• [13] J.Guillera,Ramanujanserieswithshift,J.Aust.Math.Soc.,2019inpress.
• [14] J. Guillera, A family of Ramanujan-Orr formulas for 1/π, Integral Transforms Spec. Funct., v.26, no.7, 2015, 531-538.
• [15] J.Guillera,Ramanujanseriesupside-down,J.Aust.Math.Soc.,v.97,no.1,2014,78-106.
• [16] J.Guillera,MorehypergeometricidentitiesrelatedtoRamanujan-typeseries,RamanujanJ.,v.32,no.1,2013, 5-22.
• [17] J.Guillera,AnewRamanujan-likeseriesfor1/π2,RamanujanJ.v.26,no.3,2011,369-374.
• [18] J.Guillera,OnWZ-pairswhichproveRamanujanseries,RamanujanJ.,v.22,no.3,2010,249-259.
• [19] J.Guillera,Hypergeometricidentitiesfor10extendedRamanujan-typeseries,RamanujanJ.,v.15,no.2,2008, 219-234.
• [20] J.Guillera,SomebinomialseriesobtainedbytheWZ-method,Adv.inAppl.Math.,v.29,no.4,2002,599-603.
• [21] J.Guillera,AmethodforprovingRamanujan’sseriesfor1/π,RamanujanJ.,2019,inpress.
• [22] Z-GLiu,SummationformulaandRamanujantypeseries,J.Math.Anal.Appl.,v.389,no.2,2012,1059-1065.
• [23] Z-GLiu,GausssummationandRamanujan-typeseriesfor1/π,Int.J.NumberTheory,v.8,no.2,2012,289-297.
• [24] M.Petkovšek,H.S.WilfandD.Zeilberger,A=B,A.K.Peters,Ltd.,Wellesley,Mass.,1996.
• [25] S.Ramanujan,Modularequationsandapproximationstoπ,Quart.J.Math(Oxford)45(1914)350-372.
• [26] H.M.Srivastava,J.Jhoi,Zetaandq-zetaFunctionsandAssociatedSeriesandIntegrals,Elsevier,2012.
• [27] H.S.WilfandD.Zeilberger,Rationalfunctionscertifycombinatorialidentities,J.Amer.Math.Soc.3(1990), 147-158.