Some double-angle formulas related to a generalized lemniscate function

In this paper, we will establish some double-angle formulas related to the inverse function of ∫0xdt/1-t6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _0^x \text {d}t/\sqrt{1-t^6}$$\end{document}. This function appears in Ramanujan’s Notebooks and is regarded as a generalized version of the lemniscate function.

We extend sin p,q x to (π p,q /2, π p,q ] by sin p,q (π p,q − x) and to the whole real line R as the odd 2π p,q -periodic continuation of the function. Since sin p,q x ∈ C 1 (R), we also define cos p,q x by cos p,q x := (d/dx)(sin p,q x). Then, it follows that | cos p,q x| p + | sin p,q x| q = 1.
In case ( p, q) = (2, 2), it is obvious that sin p,q x, cos p,q x and π p,q are reduced to the ordinary sin x, cos x and π , respectively. This is a reason why these functions and the constant are called generalized trigonometric functions (with parameter ( p, q)) and the generalized π , respectively.
The generalized trigonometric functions are well studied in the context of nonlinear differential equations (see [4,6,7] and the references given there). Suppose that u is a solution of the initial value problem of the p-Laplacian which is reduced to the equation −u = u of simple harmonic motion for u = sin x in case ( p, q) = (2, 2). Then, Therefore, |u | p + |u| q = 1. It is possible to show that u coincides with sin p,q defined as above. The generalized trigonometric functions are often applied to the eigenvalue problem of the p-Laplacian. Now, we are interested in finding double-angle formulas for generalized trigonometric functions. It is possible to discuss addition formulas for these functions: for instance sin 2,6 has the addition formula (3) with (2) below (see also [5] for sin 4/3,4 ), but for simplicity we will not develop this point here.
We have known the double-angle formulas of sin 2,q , sin q * ,q , and sin q * ,2 for q = 2, 3, 4 except for sin 3/2,2 , where q * := q/(q − 1) ( Table 1). For details for each formula, we refer the reader to [8] (after having proved the formula for sin 2,3 in the co-authored paper [8], the author noticed that the formula has already been obtained as "ϕ(2s)" by Cox and Shurman [3, p. 697]). It is worth pointing out that Lemma 3.1 (resp. Lemma 3.2) below connects the parameter (2, q) to (q * , q) (resp. (q * , 2)) and yields the possibility to obtain the other formula from one formula. Indeed, in this way, the formulas of sin 4/3,4 and sin 4/3,2 follow from that of sin 2,4 ([10, Subsect. 3.1] and [8, Theorem 1.1], respectively), and the formula of sin 2,3 follows from that of sin 3/2,3 ([8, Theorem 1.2]). Nevertheless, the parameter (3/2, 2) is still open because of the difficulty of the inverse problem corresponding to (10). Table 1 The parameters for which the double-angle formulas have been obtained In this paper, we wish to investigate the double-angle formula of the function sin 2,6 x, whose inverse function is defined as The function sin 2,6 x appears as the inverse of "H (v)" in Ramanujan's Notebooks [1, p. 246] and is regarded as a generalized version of the lemniscate function sin 2,4 x.
To show Theorem 1.2, the following lemma is useful.