Hypergeometric Identities Related to Roberts Reductions of Hyperelliptic Integrals
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Hypergeometric Identities Related to Roberts Reductions of Hyperelliptic Integrals Santosh B. Joshi and Daniele Ritelli Abstract. In this article starting from some reductions of hyperelliptic integrals of genus 3 into elliptic integrals, due to Michael Roberts (A Tract on the addition of Elliptic and hyperelliptic integrals, Hodger, Foster and Co, 1871) we obtain several identities which, to the best of our knowledge, are all new. The strategy used at this purpose is to evaluate Roberts integrals, in two different ways, on one side by means of elliptic integrals, obtained from the Roberts method of reduction and, on the other side, using multivariate hypergeometric functions. Mathematics Subject Classification. 33C65, 33E05, 33C05. Keywords. Lauricella function, complete elliptic integral of first and second kind, Appell function, Gauss hypergeometric, Cauchy–Schl¨ omilch transformation.
1. Introduction Pherhaps the most famous reduction scheme for hyperllelliptic integrals is due to Jacobi, [7] where the following identity, being a > b > 1 and c = √ √ −( a − b)2 /(1 − a)(1 − b) is established √ 1 ab + z dz z(z − 1)(z − ab)(z − a)(z − b) 0 1 dx 1 (1.1) = (1 − a)(1 − b) 0 x(1 − x)(1 − cx) 0123456789().: V,-vol
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S. B. Joshi and D. Ritelli
Results Math
Identity (1.1) stems from the second degree transformation x = (1 − a)(1 − b)z/(z − a)(z − b). Generally speaking, reductions of hyperelliptic integrals are possible when a variable transformation links an hyperelliptic integral to an elliptic integral. Implications and conseguences of this situation has been exploited, for example, in the articles [11–13], in which historically famous reductions due to the aforementioned Jacobi [7], Legendre [9,10], Hermite [6] and Goursat [4] have been used, to obtain, several identities between hypergeometric functions with different numbers of variables. In this article, this technique is used starting from a class of reductions, probably less known, due to the Irish mathematician Micheal Roberts (1817– 1882). A scientific biography of Roberts is provided by [18]. In [15], section 63, Roberts presented reductions of some genus 3 hyperelliptic integrals of the form: xn xn dx = dx, (1.2) Rn = P (x) x8 − px6 + qx4 − px2 + 1 with n non negative even integer, dealing in particular with exponents n = 0, 2, 4. Reduction is achieved using the second degree Cauchy–Schl¨ omilch transformation: u=x +
1 u 1 1 ⇐⇒ x = u ± u2 − 4 =⇒ dx = 1± √ du. (1.3) x 2 2 u2 − 4
For hystorical references about Cauchy–Schl¨ omilch transformation, it is noteworthy the recent paper [1], while in [12] the use of the transformation by Legendre, very similar to the one of Roberts, is mentioned. The double sign depends on the non monotonicity of the transformation (1.3): sign + has to be taken when x ∈ [0, 1] and − when x ∈ (1, ∞) according to Fig. 1: Transformation (1.3) generates two integrals: n−1 √ 2 u −4±u 1 1 √ du. (1.4) Rn = n−1 2 u2 − 4 u4 − (p + 4)u2 + q + 2p + 2 √ 2 The √ first inte
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