@article{Shynarbek2026, 
author = {Nurdaulet Shynarbek and Shirali Kadyrov and Alibek Orynbassar and Muhammad Ateeq Tahir},
title = {Raising operators and a parametric polynomial continued fraction for        π    2},
year = {2026},
journal = {Electronic Research Archive},
volume = {34},
number = {6},
pages = {3895-3913},
keywords = {continued fractions, polynomial continued fractions, J-fractions, parameter-raising operators, central binomial coefficients, telescoping, π2},
url = {https://www.sciopen.com/article/10.3934/era.2026175},
doi = {10.3934/era.2026175},
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      /    (      n    2              (                      2        n            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  )  =                    (                              2          u                u                    )              3              π      2        18    +      ρ    u    ,        ρ    u    ∈      Q    .Thus, the operator identity and the telescoping identity provide the algebraic mechanism behind the evaluation of the whole family.}
}