Raising operators and a parametric polynomial continued fraction for $ \pi^2 $

Nurdaulet Shynarbek; Shirali Kadyrov; Alibek Orynbassar; Muhammad Ateeq Tahir; Nurdaulet Shynarbek; Shirali Kadyrov; Alibek Orynbassar; Muhammad Ateeq Tahir

doi:10.3934/era.2026175

Electronic Research Archive

2026, Volume 34, Issue 6: 3895-3913. doi: 10.3934/era.2026175

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Raising operators and a parametric polynomial continued fraction for $ \pi^2 $

1.
Department of Pedagogy of Natural Sciences, SDU University, Kaskelen 040900, Kazakhstan
2.
Department of General Education, New Uzbekistan University, Tashkent 100008, Uzbekistan
3.
School of Digital Technologies, Narxoz University, Almaty 050035, Kazakhstan

Received: 26 March 2026 Revised: 30 April 2026 Accepted: 07 May 2026 Published: 12 May 2026

We study a one-parameter family of polynomial $ J $-fractions whose coefficients depend polynomially on the index and on an integer parameter $ u\ge0 $. After a factorial normalization of the denominator sequence, the difference of two consecutive convergents factors into the fixed central-binomial Apéry-type term $ 1/(n^2\binom{2n}{n}) $ 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{2u}{u}^3\frac{\pi^2}{18}+\rho_u, \qquad \rho_u\in \mathbb{Q} . $
Thus, the operator identity and the telescoping identity provide the algebraic mechanism behind the evaluation of the whole family.
- continued fractions,
- polynomial continued fractions,
- $ J $-fractions,
- parameter-raising operators,
- central binomial coefficients,
- telescoping,
- $ \pi^2 $
Citation: Nurdaulet Shynarbek, Shirali Kadyrov, Alibek Orynbassar, Muhammad Ateeq Tahir. Raising operators and a parametric polynomial continued fraction for $ \pi^2 $[J]. Electronic Research Archive, 2026, 34(6): 3895-3913. doi: 10.3934/era.2026175

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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\ge0 $. After a factorial normalization of the denominator sequence, the difference of two consecutive convergents factors into the fixed central-binomial Apéry-type term $ 1/(n^2\binom{2n}{n}) $ 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{2u}{u}^3\frac{\pi^2}{18}+\rho_u, \qquad \rho_u\in \mathbb{Q} . $

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

References

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