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Cyclic Vanishing Identities of Sun-Pan Type: Analytic and Modular Perspectives

Ken Nagai

TL;DR

The paper investigates cyclic vanishing identities of Sun–Pan type for Bernoulli polynomials and their $q$-analogues. It develops a twofold framework: minimal Appell axioms that enforce cyclic vanishing, and a modular perspective via period polynomials and the modular relation $(ST)^3=-I$, showing the same identities arise from analytic and arithmetic structures. It unifies analysis, combinatorics, and arithmetic geometry, extending to $q$- and elliptic analogues and connecting to polylogarithms, $L$-values, and motivic theory. The results clarify the structural origin of Sun–Pan identities and suggest deeper links to mixed Tate motives and modular forms.

Abstract

We revisit the cyclic identities of Sun--Pan type for Bernoulli polynomials and their $q$-analogues. From the analytic side, we formulate minimal Appell axioms that force cyclic vanishing identities, extending naturally to $q$-Appell sequences and analytic Bernoulli functions. From the modular side, we show that the same relations arise as period polynomial identities associated with Eisenstein series, reflecting the symmetry $(ST)^3=-I$ of the modular group. These two complementary perspectives place the Sun--Pan cyclic identities at the crossroads of number theory, special functions, and modular forms, and suggest further connections to polylogarithms, $L$-values, and mixed Tate motives.

Cyclic Vanishing Identities of Sun-Pan Type: Analytic and Modular Perspectives

TL;DR

The paper investigates cyclic vanishing identities of Sun–Pan type for Bernoulli polynomials and their -analogues. It develops a twofold framework: minimal Appell axioms that enforce cyclic vanishing, and a modular perspective via period polynomials and the modular relation , showing the same identities arise from analytic and arithmetic structures. It unifies analysis, combinatorics, and arithmetic geometry, extending to - and elliptic analogues and connecting to polylogarithms, -values, and motivic theory. The results clarify the structural origin of Sun–Pan identities and suggest deeper links to mixed Tate motives and modular forms.

Abstract

We revisit the cyclic identities of Sun--Pan type for Bernoulli polynomials and their -analogues. From the analytic side, we formulate minimal Appell axioms that force cyclic vanishing identities, extending naturally to -Appell sequences and analytic Bernoulli functions. From the modular side, we show that the same relations arise as period polynomial identities associated with Eisenstein series, reflecting the symmetry of the modular group. These two complementary perspectives place the Sun--Pan cyclic identities at the crossroads of number theory, special functions, and modular forms, and suggest further connections to polylogarithms, -values, and mixed Tate motives.

Paper Structure

This paper contains 4 sections, 5 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Minimal Axioms and Cyclic Vanishing
  3. Modular Origin of Cyclic Identities
  4. Concluding Remarks

Key Result

Theorem 2.1

Let $\{F_n(x)\}$ satisfy axioms (1)--(3). For integers $r+s+t=n$ and $x+y+z=1$, one has where

Theorems & Definitions (9)

  • Theorem 2.1: Classical cyclic identities
  • proof
  • Definition 2.2: $q$-cyclic bracket
  • Theorem 2.3: $q$-cyclic identity
  • proof
  • Corollary 2.4: q-Bernoulli and q-Euler polynomials
  • proof
  • Theorem 3.1: Three-term relation
  • Corollary 3.2: Sun--Pan identity