# Jacob's ladders and completely new exact synergetic formula for Jacobi's elliptic functions together with Bessel's functions excited by the function $|\zeta(1/2+it)|^2$

## TL;DR

This paper introduces new transmutations and synergetic formulas involving Jacobi's elliptic functions, Bessel's functions, and the Riemann zeta function, revealing complex interactions and exact metafunctional equations.

## Contribution

It presents five new transmutations of a core formula and ten exact metafunctional equations, advancing the understanding of interactions among classical special functions.

## Key findings

- Five new transmutations of the mother formula
- Ten exact metafunctional equations
- Revealed synergetic interactions between functions

## Abstract

In this paper we obtain a set of five new transmutations of the mother formula. Further, we obtain the second set of ten exact metafunctional equations by crossbreeding on every two elements of the previous set. Elements of the last set represent a kind of synergetic formulae describing cooperative interactions between corresponding sets of values of basic classical functions.