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New analogues of Clausen’s identities arising from the theory of modular forms. (English) Zbl 1234.33009

The authors prove three “Clausen-type” identities, a representative example of which is \[ \left(\sum_{n \geq 0} r_nx^n \right)^2 = \frac{1}{1+8x^2}\sum_{n \geq 0} \binom{2n}{n}r_n\left(\frac{x(1+x)(1-8x)}{(1+8x^2)^2}\right)^n, \] where \(r_n\) is defined by \[ r_n := \sum_{k=0}^n \binom{n}{k}^3. \] They carefully explain how to prove their identities using the theory of modular forms, and how, in this setting, the identities may be associated with Ramaujan-type series for \(1/\pi\). One example is \[ \frac{25}{2\pi} = \sum_{n \geq 0} (9n+2)\binom{2n}{n}r_n\left(\frac{1}{50}\right)^n. \]

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
05A10 Factorials, binomial coefficients, combinatorial functions
11F03 Modular and automorphic functions
11F11 Holomorphic modular forms of integral weight
11Y60 Evaluation of number-theoretic constants

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