[Paper Review] An Apéry-like difference equation for Catalan's constant

Wadim Zudilin|ArXiv.org|Jan 4, 2002

Advanced Differential Equations and Dynamical Systems2 references4 citations

TL;DR

This paper constructs a second-order linear difference equation with rational coefficients that generates sequences approximating Catalan's constant G. Using hypergeometric series and integral representations, it proves that the ratio of solutions converges to G and establishes nearly integral properties for the sequences, providing a new continued fraction expansion and integral formula for linear forms in G.

ABSTRACT

Applying Zeilberger's algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a second-order difference equation for these forms and their coefficients. As a consequence we obtain a new way of fast calculation of Catalan's constant as well as a new continued-fraction expansion for it. Similar arguments can be put forward to indicate a second-order difference equation and a new continued fraction for $ζ(4)=π^4/90$, and we announce corresponding results at the end of this paper.

Motivation & Objective

• To construct a second-order linear recurrence with rational coefficients whose solution ratio converges to Catalan’s constant G.
• To establish arithmetic properties (nearly integral forms) of the recurrence sequences to support irrationality proofs.
• To derive a new continued fraction expansion for G using the recurrence’s convergents.
• To provide a double integral representation for the linear forms $ u_n G - v_n $, analogous to Beukers’ work on $ \zeta(2) $.

Proposed method

• Derives a new second-order difference equation with polynomial coefficients $ p(n) = 20n^2 - 8n + 1 $ and $ q(n) = 3520n^6 + 5632n^5 + 2064n^4 - 384n^3 - 156n^2 + 16n + 7 $, governing sequences $ u_n $ and $ v_n $.
• Uses a very-well-poised hypergeometric series $ F_n $ defined via a rational function $ R_n(t) $, which is shown to equal a combination of Dirichlet beta values.
• Applies Euler-type integral representations to express $ u_n G - v_n $ as a double integral over $[0,1]^2$ with a rational kernel involving $ (1 - xy)^{-(n+1)} $.
• Applies Poincaré’s theorem to the recurrence to derive asymptotic growth rates of $ u_n $ and $ v_n $, showing $ |u_n G - v_n|^{1/n} \to \left| \frac{1 - \sqrt{5}}{2} \right|^5 $.
• Transforms the recurrence’s convergents into a continued fraction expansion for G with terms involving $ (2n-1)^4 (2n)^4 p(n-1)p(n+1) $ and $ q(n) $.
• Establishes that $ 2^{4n+3} D_n u_n \in \mathbb{Z} $ and $ 2^{4n+3} D_{2n-1}^3 v_n \in \mathbb{Z} $, indicating nearly integral behavior.

Experimental results

Research questions

• RQ1 Can a recurrence relation be constructed for Catalan’s constant analogous to Apéry’s proof for $ \zeta(2) $ and $ \zeta(3) $?
• RQ2 Do the solutions of the recurrence yield linear forms in $ 1 $ and $ G $ with nearly integral coefficients and rapid convergence?
• RQ3 Can a double integral representation be derived for the error $ u_n G - v_n $, similar to Beukers’ formula for $ \zeta(2) $?
• RQ4 Does the recurrence generate a continued fraction expansion for $ G $, and what is its structure?
• RQ5 Can the nearly integral properties of the sequences support a proof of the irrationality of $ G $?

Key findings

• The recurrence $ (2n+1)^2(2n+2)^2 p(n) u_{n+1} - q(n) u_n - (2n-1)^2(2n)^2 p(n+1) u_{n-1} = 0 $ with $ p(n) = 20n^2 - 8n + 1 $ and $ q(n) $ as given generates sequences $ u_n $, $ v_n $ such that $ \lim_{n \to \infty} v_n / u_n = G $.
• The sequences satisfy $ 2^{4n+3} D_n u_n \in \mathbb{Z} $ and $ 2^{4n+3} D_{2n-1}^3 v_n \in \mathbb{Z} $, indicating strong arithmetic control.
• The error term satisfies $ \lim_{n \to \infty} |u_n G - v_n|^{1/n} = \left| \frac{1 - \sqrt{5}}{2} \right|^5 \approx e^{-2.406} < 1 $, confirming rapid convergence.
• A new continued fraction expansion for $ G $ is derived with partial numerators $ (2n-1)^4 (2n)^4 p(n-1)p(n+1) $ and denominators $ q(n) $, valid for all $ n \geq 0 $.
• The linear form $ u_n G - v_n $ admits the double integral representation $ \frac{(-1)^n}{4} \int_0^1 \int_0^1 \frac{x^{n-1/2}(1-x)^n y^n (1-y)^{n-1/2}}{(1 - xy)^{n+1}} \, dx \, dy $.
• Despite strong arithmetic properties, the linear forms $ 2^{4n} D_{2n-1}^2 (u_n G - v_n) $ do not tend to zero, so the method does not prove irrationality of $ G $.