[Paper Review] Higher Apery-like numbers arising from special values of the spectral zeta function for the non-commutative harmonic oscillator

Kazufumi Kimoto|ArXiv.org|Jan 6, 2009

Quantum Mechanics and Non-Hermitian Physics20 references4 citations

TL;DR

This paper introduces higher Apéry-like numbers $ J_k(n) $ arising from special values of the spectral zeta function $ \zeta_Q(s) $ for the non-commutative harmonic oscillator. It establishes recurrence relations connecting $ J_k(n) $ and $ J_{k-2}(n) $, derives generating functions satisfying singly confluent Heun equations, and defines normalized rational parts $ \tilde{J}_k(n) $, proving congruences analogous to those of classical Apéry numbers, including $ \tilde{J}_2((p-1)/2) \equiv A_2((p-1)/2) \pmod{p^2} $ and conjectured supercongruences modulo $ p^3 $.

ABSTRACT

A generalization of the Apery-like numbers, which is used to describe the special values $ζ_Q(2)$ and $ζ_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a recurrence relation for them, which shows a ladder structure among them. Further, we consider the `rational part' of the higher Apery-like numbers. We discuss several kinds of congruence relations among them, which are regarded as an analogue of the ones among Apery numbers.

Motivation & Objective

• To generalize Apéry-like numbers to higher orders $ k \geq 2 $, linking them to special values $ \zeta_Q(k) $ of the spectral zeta function for the non-commutative harmonic oscillator.
• To establish a recurrence structure connecting $ J_k(n) $ and $ J_{k-2}(n) $, revealing a ladder-like hierarchy among the numbers.
• To define normalized rational parts $ \tilde{J}_k(n) $ via change of variables in the differential equation, enabling congruence analysis.
• To investigate congruence relations among $ \tilde{J}_k(n) $, drawing analogies to classical Apéry number congruences.
• To formulate and support conjectures on supercongruences, including modulo $ p^3 $ and $ p^r $, extending known results to higher $ k $.

Proposed method

• Define higher Apéry-like numbers $ J_k(n) $ via a $ k $-fold integral over $ [0,1]^k $, involving rational functions of $ x_1, \dots, x_k $, with a weight factor depending on $ n $.
• Derive a three-term inhomogeneous recurrence relation for $ J_k(n) $, which translates into a singly confluent Heun differential equation for the generating function.
• Establish a recursive ladder structure by relating $ J_k(n) $ to $ J_{k-2}(n) $, suggesting a potential modular or arithmetic link between $ \zeta_Q(k) $ and $ \zeta_Q(k-2) $.
• Introduce normalized rational parts $ \tilde{J}_k(n) $ by a change of variable in the differential equation, such that $ J_k(n) $ is a linear combination of Riemann zeta values with coefficients $ \tilde{J}_m(n) $.
• Use $ p $-adic analysis and binomial coefficient identities modulo $ p^2 $ and $ p^3 $ to prove congruences, including $ \tilde{J}_2((p-1)/2) \equiv A_2((p-1)/2) \pmod{p^2} $.
• Formulate and support conjectures on higher-order supercongruences, such as $ \sum_{n=0}^{p-1} \tilde{J}_2(n)^2 \equiv \left(\frac{-1}{p}\right) \pmod{p^3} $, and $ \tilde{J}_2(mp^r - 1) \equiv \left(\frac{-1}{p}\right) \tilde{J}_2(mp^{r-1} - 1) \pmod{p^r} $.

Experimental results

Research questions

• RQ1 How do higher Apéry-like numbers $ J_k(n) $ for $ k \geq 2 $ arise from the special values $ \zeta_Q(k) $ of the spectral zeta function of the non-commutative harmonic oscillator?
• RQ2 What recurrence structure connects $ J_k(n) $ and $ J_{k-2}(n) $, and how does this reflect a deeper arithmetic or spectral property of $ \zeta_Q(s) $?
• RQ3 What is the role of the normalized rational parts $ \tilde{J}_k(n) $, and how do they facilitate the study of $ p $-adic congruences?
• RQ4 Are there supercongruences analogous to those of classical Apéry numbers, such as $ \sum_{n=0}^{p-1} \tilde{J}_2(n)^2 \equiv \left(\frac{-1}{p}\right) \pmod{p^3} $, and how do they generalize to higher $ k $?
• RQ5 Can the congruence $ \tilde{J}_2(mp^r - 1) \equiv \left(\frac{-1}{p}\right) \tilde{J}_2(mp^{r-1} - 1) \pmod{p^r} $ be extended or proven for higher $ k $, and what does it imply about the arithmetic nature of $ \zeta_Q(k) $?

Key findings

• The higher Apéry-like numbers $ J_k(n) $ satisfy a three-term inhomogeneous recurrence relation that links $ J_k(n) $ to $ J_{k-2}(n) $, indicating a ladder structure across $ k $.
• The generating function of $ J_k(n) $ satisfies a singly confluent Heun differential equation, which arises from the recurrence and reflects the spectral zeta function's analytic structure.
• The normalized rational parts $ \tilde{J}_k(n) $ are defined via a change of variable in the differential equation, allowing $ J_k(n) $ to be expressed as a linear combination of $ \zeta(k), \zeta(k-2), \dots $ with coefficients $ \tilde{J}_m(n) $.
• It is proven that $ \tilde{J}_2((p-1)/2) \equiv A_2((p-1)/2) \pmod{p^2} $, where $ A_2(n) $ is the classical Apéry number for $ \zeta(2) $, establishing a direct link between normalized higher and classical Apéry numbers.
• The paper conjectures that $ \sum_{n=0}^{p-1} \tilde{J}_2(n)^2 \equiv \left(\frac{-1}{p}\right) \pmod{p^3} $, a supercongruence analogous to Rodriguez-Villegas-type results.
• A deeper conjecture states that $ \tilde{J}_2(mp^r - 1) \equiv \left(\frac{-1}{p}\right) \tilde{J}_2(mp^{r-1} - 1) \pmod{p^r} $ for all $ m, r \geq 1 $, and that $ \tilde{J}_2((mp^r - 1)/2) $ satisfies a three-term congruence involving $ \lambda_p $, the Fourier coefficient of a modular form of weight 6.