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Properties of generalized Jacobi elliptic functions with three parameters

Hajime Sato, Nagi Suzuki, Shingo Takeuchi

TL;DR

This work develops a three-parameter generalization of Jacobi elliptic functions and generalized complete elliptic integrals, linking them to nonlinear ODEs involving the $p$-Laplacian. It derives Wallis-type integral formulae for powers of these generalized functions, a Legendre-type relation for the generalized integrals (equivalent to Elliott’s hypergeometric identity), and nontrivial binomial inequalities, with reductions to classical cases when the parameters specialize. The results provide explicit hypergeometric representations, recurrence relations, and Wronskian-based proofs, unifying classical elliptic function theory with three-parameter generalizations. These developments extend analytic tools for nonlinear differential equations and hypergeometric function theory, with potential applications in mathematical physics and applied analysis.

Abstract

Jacobi elliptic functions and complete elliptic integrals are generalized using three parameters. These generalized functions and integrals are closely related to ordinary differential equations involving $p$-Laplacian. In this paper, Wallis-type integral formulae are constructed for the generalized Jacobi elliptic functions. Moreover, for the generalized complete elliptic integrals, a Legendre-type relation is derived, which is equivalent to Elliott's identity for Gaussian hypergeometric series, along with its implications. In addition, nontrivial inequalities on binomial expansions of generalized Jacobi elliptic functions are given.

Properties of generalized Jacobi elliptic functions with three parameters

TL;DR

This work develops a three-parameter generalization of Jacobi elliptic functions and generalized complete elliptic integrals, linking them to nonlinear ODEs involving the -Laplacian. It derives Wallis-type integral formulae for powers of these generalized functions, a Legendre-type relation for the generalized integrals (equivalent to Elliott’s hypergeometric identity), and nontrivial binomial inequalities, with reductions to classical cases when the parameters specialize. The results provide explicit hypergeometric representations, recurrence relations, and Wronskian-based proofs, unifying classical elliptic function theory with three-parameter generalizations. These developments extend analytic tools for nonlinear differential equations and hypergeometric function theory, with potential applications in mathematical physics and applied analysis.

Abstract

Jacobi elliptic functions and complete elliptic integrals are generalized using three parameters. These generalized functions and integrals are closely related to ordinary differential equations involving -Laplacian. In this paper, Wallis-type integral formulae are constructed for the generalized Jacobi elliptic functions. Moreover, for the generalized complete elliptic integrals, a Legendre-type relation is derived, which is equivalent to Elliott's identity for Gaussian hypergeometric series, along with its implications. In addition, nontrivial inequalities on binomial expansions of generalized Jacobi elliptic functions are given.

Paper Structure

This paper contains 6 sections, 20 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Differential equations for GJEFs
  4. Integral formulae
  5. Legendre-type relation
  6. Inequalities on binomial expansions

Key Result

Theorem 3.1

$(\mathrm{i})$ For $x \in \mathbb{R}$, $y=\operatorname{sn}_{p,q,r}{x}$ satisfies In particular, $y=\operatorname{sn}_{p,q,p^*}x$ satisfies the Allen-Cahn-type equation$:$ $(\mathrm{ii})$ For $x \in \mathbb{R}$, $y=\phi_p(\operatorname{cn}_{p,q,r}{x})$ satisfies In particular, $y=\phi_p(\operatorname{cn}_{p,q,q}{x})$ satisfies $(\mathrm{iii})$ For $x \in \mathbb{R}$, $y=\phi_{r^*}(\operatorname

Theorems & Definitions (42)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Corollary 4.2: Kobayashi-Takeuchi
  • proof
  • Theorem 4.3: Takeuchi2016b
  • proof
  • Theorem 4.4: Wallis-type integral formula I
  • ...and 32 more