## 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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