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The holonomic triangle: from a symmetry between $e$ and $π$ to additive Gamma functions

Benoit Cloitre

TL;DR

The paper develops the additive Gamma function (AGF) framework to unify discrete holonomic recurrences with continuous meromorphic structures via additive functional equations (AFEs). It proves an equivalence: a meromorphic function is an AGF of order $r$ iff it arises as the connection constant of a holonomic sequence in $(n,z)$ with integer-slope asymptotics, linking $u_n(z)\sim \Lambda(n,z)h(z)$ to a rational-coefficient AFE for $h$. Two archetypes are analyzed in depth: an irregular case $f(z)$ tied to incomplete Gamma for the $\mathrm{e}$-world, and a regular case $g(z)$ expressed as Gamma ratios for the $\pi$-world; together with Euler’s $\Gamma(z)$ as the order-1 prototype, they populate an AGF hierarchy. The work introduces the Integer Slope Condition, develops a holonomic triangle connecting P-recurrences, AFEs, and D-finite ODEs, and provides explicit holonomic certificates, offering a unifying lens for discrete-continuous dualities in asymptotics and special functions.

Abstract

Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $π$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational coefficients. We call such functions additive Gamma functions (AGFs), recognizing Euler's $Γ(z)$ as the order-1 prototype. Our theory reveals a structural dichotomy: one AGF is expressible as Gamma ratios (regular case), another involves incomplete Gamma (irregular case). AGFs complete a holonomic triangle between P-recursive sequences, additive functional equations, and differential equations, unifying discrete and continuous perspectives under the condition that Gamma factors in asymptotics have integer slopes.

The holonomic triangle: from a symmetry between $e$ and $π$ to additive Gamma functions

TL;DR

The paper develops the additive Gamma function (AGF) framework to unify discrete holonomic recurrences with continuous meromorphic structures via additive functional equations (AFEs). It proves an equivalence: a meromorphic function is an AGF of order iff it arises as the connection constant of a holonomic sequence in with integer-slope asymptotics, linking to a rational-coefficient AFE for . Two archetypes are analyzed in depth: an irregular case tied to incomplete Gamma for the -world, and a regular case expressed as Gamma ratios for the -world; together with Euler’s as the order-1 prototype, they populate an AGF hierarchy. The work introduces the Integer Slope Condition, develops a holonomic triangle connecting P-recurrences, AFEs, and D-finite ODEs, and provides explicit holonomic certificates, offering a unifying lens for discrete-continuous dualities in asymptotics and special functions.

Abstract

Two linear recurrences exhibit mirror symmetry connecting the constants and . When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational coefficients. We call such functions additive Gamma functions (AGFs), recognizing Euler's as the order-1 prototype. Our theory reveals a structural dichotomy: one AGF is expressible as Gamma ratios (regular case), another involves incomplete Gamma (irregular case). AGFs complete a holonomic triangle between P-recursive sequences, additive functional equations, and differential equations, unifying discrete and continuous perspectives under the condition that Gamma factors in asymptotics have integer slopes.
Paper Structure (42 sections, 13 theorems, 119 equations, 2 figures)

This paper contains 42 sections, 13 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The mirror symmetry
  3. The arithmetic duality
  4. From arithmetic linear forms to meromorphic functions
  5. Main results
  6. Uniqueness and the regular/irregular dichotomy.
  7. Related work and positioning
  8. Organization
  9. Technical preliminaries
  10. Asymptotic analysis: the world of $\mathrm{e}$
  11. Generating function and differential equation
  12. Singularity analysis and asymptotic behavior
  13. Discrete recurrence and arithmetic structure
  14. Analytic continuation and functional equation
  15. Explicit formula
  16. ...and 27 more sections

Key Result

Theorem 1.3

For the parametrized sequence $u_n(m)$ defined by eq:param, there exists a connection constant $f(m)$ such that The function $f(m)$ extends to a meromorphic function $f(z)$ on $\mathbb{C}$ satisfying the additive functional equation The function $f(z)$ admits an explicit formula in terms of the incomplete Gamma function $\gamma(a,x)$ and the confluent hypergeometric function ${}_1F_1$ (see eq:f-

Figures (2)

  • Figure 1: The holonomic triangle. The vertices represent the P-recurrence (order $r$), the AFE (order $r$), and the D-finite ODE (order $\leq (d+1)r$ for the ordinary generating function, where $d$ is the maximum degree of polynomial coefficients in the P-recurrence). Note: the bound $(d+1)r$ is worst-case; in practice the ODE order is often much lower---for instance, our order-2 examples yield ODEs of order 1.
  • Figure 2: The holonomic triangle for Euler's Gamma function, illustrating the three-way equivalence between P-recurrence, AFE, and D-finite ODE under the integer slope condition.

Theorems & Definitions (24)

  • Definition 1.1: Additive Gamma function
  • Remark 1.2
  • Theorem 1.3: Connection constant in the world of $\mathrm{e}$
  • Theorem 1.4: Connection constant in the world of $\pi$
  • Theorem 2.1: Carlson
  • Lemma 2.2: Uniqueness under AFE
  • proof
  • Remark 3.1: Order of the ODE
  • Lemma 3.2: Growth bounds for $f$
  • Proposition 4.1: Alternation and closed formulas
  • ...and 14 more