Raising operators and a parametric polynomial continued fraction for π²

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Abstract

We study a one-parameter family of polynomial J-fractions whose coefficients depend polynomially on the index and on an integer parameter u ≥ 0. After a factorial normalization of the denominator sequence, the difference of two consecutive convergents factors into the fixed central-binomial Apéry-type term $1 /\left(n^{2}\binom{2 n}{n}\right)$ and a rational factor determined by a normalized polynomial family P_u. We construct P_u by an explicit parameter-raising operator. We then prove a parameter-shift telescoping identity, which allows induction on u and gives

$X(u)=\binom{2 u}{u}^{3} \frac{\pi^{2}}{18}+\rho_{u}, \quad \rho_{u} \in \mathbb{Q} .$

Thus, the operator identity and the telescoping identity provide the algebraic mechanism behind the evaluation of the whole family.

Keywords

continued fractions / polynomial continued fractions / J-fractions / parameter-raising operators / central binomial coefficients / telescoping / π²

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References

[1] T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse: Math. Ser. 1, 8 (1894), J1-J122. https://doi.org/10.5802/afst.108

[2] O. Perron, Die Lehre von den Kettenbrüchen , Teubner, Stuttgart, 1954.

[3] H. S. Wall, Analytic Theory of Continued Fractions , New York, 1948.

[4] W. B. Jones, W. J. Thron, Continued Fractions: Analytic Theory and Applications , Addison-Wesley, Reading, MA, 1980.

[5] L. Lorentzen, H. Waadeland, Continued Fractions with Applications , Atlantis Press, Paris, 2008. https://doi.org/10.2991/978-94-91216-37-4

[6] A. Cuyt, V. B. Petersen, B. Verdonk, H. Waadeland, W. B. Jones, Handbook of Continued Fractions for Special Functions , Springer, Dordrecht, 2008. https://doi.org/10.1007/978-1-4020-6949-9

[7] P. Flajolet, Combinatorial aspects of continued fractions, Discrete Math., 32 (1980), 125-161. https://doi.org/10.1016/0012-365X(80)90050-3

[8] D. Bowman, J. McLaughlin, Polynomial continued fractions, preprint, arXiv:1812.08251.

[9] R. Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque, 61 (1979), 11-13.

[10] F. Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc., 11 (1979), 268-272. https://doi.org/10.1112/blms/11.3.268

[11] A. Poorten, R. Apéry A proof that Euler missed ..., Math. Intell., 1 (1979), 195-203. https://doi.org/10.1007/BF03028234

[12] J. M. Borwein, R. Girgensohn, Evaluations of binomial series, Aequ. Math., 70 (2005), 25-36. https://doi.org/10.1007/s00010-005-2774-x

[13] H. Cohen, Parametric continued fractions for π², ζ(3), and other constants , preprint, arXiv:2304.11727.

[14] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B , New York, 1996. https://doi.org/10.1201/9781439864500

[15] D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math., 32 (1990), 321-368. https://doi.org/10.1016/0377-0427(90)90042-X

[16] F. Chyzak, An extension of Zeilberger’s fast algorithm to general holonomic functions, Discrete Math., 217 (2000), 115-134. https://doi.org/10.1016/S0012-365X(99)00259-9

[17] C. Koutschan, Holonomic functions in Mathematica, ACM Commun. Comput. Algebra, 47 (2014), 179-182. https://doi.org/10.1145/2576802.2576831

[18] R. A. Horn, C. R. Johnson, Matrix Analysis , Cambridge University Press, Cambridge, 2013. https://doi.org/10.1017/CBO9781139020411