[Paper Review] New series for powers of $\pi$ and related congruences

Zhi‐Wei Sun|arXiv (Cornell University)|Nov 13, 2019

Advanced Mathematical Identities46 references5 citations

TL;DR

This paper presents 97 new Ramanujan-type series for powers of $/pi$ using symbolic computation, generalizing central binomial and trinomial coefficients via generalized central trinomial coefficients $T_k(b,c)$. The key contribution is a novel characterization of rational Ramanujan-type series for $1//pi$ through $p$-adic congruences, leading to 117 new conjectural series, including examples involving imaginary quadratic fields with class number 8.

ABSTRACT

Via symbolic computation we deduce 97 new type series for powers of $\\pi$ related to Ramanujan-type series. Here are three typical examples: $$\\sum_{k=0}^\\infty \\frac{P(k) \\binom{2k}k\\binom{3k}k \\binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}} =\\frac{18\ imes557403^3\\sqrt{10005}}{5\\pi}$$ with \\begin{align*}P(k) = &637379600041024803108 k^2 + 657229991696087780968 k \\\\&+ 19850391655004126179, \\end{align*} $$\\sum_{k=1}^\\infty \\frac{(3k+1)16^k}{(2k+1)^2k^3\\binom{2k}k^3} = \\frac{\\pi^2-8}2,$$ and $$\\sum_{n=0}^\\infty\\frac{3n+1}{(-100)^n} \\sum_{k=0}^n{n\\choose k}^2T_k(1,25)T_{n-k}(1,25) = \\frac{25}{8\\pi},$$ where the generalized central trinomial coefficient $T_k(b,c)$ denotes the coefficient of $x^k$ in the expansion of $(x^2+bx+c)^k$. We also formulate a general characterization of rational Ramanujan-type series for $1/\\pi$ via congruences, and pose 117 new conjectural series for powers of $\\pi$ via looking for corresponding congruences. For example, we conjecture that $$\\sum_{k=0}^\\infty\\frac{39480k+7321}{(-29700)^k}T_k(14,1)T_k(11,-11)^2=\\frac{6795\\sqrt5}{\\pi}.$$ Eighteen of the new series in this paper involve some imaginary quadratic fields with class number $8$.

Motivation & Objective

• To derive new Ramanujan-type series for powers of $\pi$ using generalized central trinomial coefficients $T_k(b,c)$.
• To establish a general characterization of rational Ramanujan-type series for $1/\pi$ via $p$-adic congruences.
• To formulate 117 new conjectural series for powers of $\pi$ by identifying corresponding $p$-adic congruences.
• To explore connections between Ramanujan-type series, binomial coefficients, and binary quadratic forms.
• To extend previous work on $p$-adic analogues of Ramanujan-type series to include generalized trinomial coefficients and new series types.

Proposed method

• Employ symbolic computation to derive new series of the form $\sum_{k=0}^{\infty} \frac{P(k) \binom{2k}{k} \binom{3k}{k} \binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}} = \frac{18 \times 557403^3 \sqrt{10005}}{5\pi}$, where $P(k)$ is a quadratic polynomial.
• Define generalized central trinomial coefficients $T_k(b,c)$ as the coefficient of $x^k$ in $(x^2 + bx + c)^k$, generalizing central binomial and trinomial coefficients.
• Utilize $p$-adic congruences of the form $\sum_{k=0}^{p-1} \frac{bk+c}{m^k} a(k) \equiv cp \left(\frac{\varepsilon_d d}{p}\right) \pmod{p^3}$ to characterize rational Ramanujan-type series.
• Apply the Laplace-Heine asymptotic formula and properties of Legendre polynomials to analyze the growth and behavior of $T_k(b,c)$.
• Use the recurrence $ (n+1)T_{n+1}(b,c) = (2n+1)bT_n(b,c) - n(b^2 - 4c)T_{n-1}(b,c) $ to compute $T_k(b,c)$ efficiently.
• Formulate conjectures based on observed $p$-adic patterns, such as $p^2 \sum_{k=1}^{p-1} \frac{(105k-44)T_{k-1}}{k^2 \binom{2k}{k}^2 3^{k-1}} \equiv 11\left(\frac{p}{3}\right) \pmod{p^2}$, and relate them to infinite series identities.

Experimental results

Research questions

• RQ1 Can new Ramanujan-type series for powers of $\pi$ be systematically derived using generalized central trinomial coefficients $T_k(b,c)$?
• RQ2 What is the general $p$-adic congruence structure that characterizes rational Ramanujan-type series for $1/\pi$?
• RQ3 How can symbolic computation be used to generate and verify new series for $\pi$ and its powers?
• RQ4 What is the connection between Ramanujan-type series and imaginary quadratic fields, particularly those with class number 8?
• RQ5 Can exact closed forms be established for certain infinite series involving $T_k(b,c)$, such as $\sum_{k=1}^{\infty} \frac{(105k-44)T_{k-1}}{k^2 \binom{2k}{k}^2 3^{k-1}}$?

Key findings

• The paper derives 97 new Ramanujan-type series for powers of $\pi$, including $\sum_{k=0}^{\infty} \frac{P(k) \binom{2k}{k} \binom{3k}{k} \binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}} = \frac{18 \times 557403^3 \sqrt{10005}}{5\pi}$ with a quadratic polynomial $P(k)$.
• Eighteen of the new series involve imaginary quadratic fields with class number 8, linking them to deep arithmetic properties.
• The paper formulates a general characterization of rational Ramanujan-type series for $1/\pi$ via $p$-adic congruences involving Legendre symbols.
• Conjecture 10.1 proposes exact evaluations: $\sum_{k=1}^{\infty} \frac{(105k-44)T_{k-1}}{k^2 \binom{2k}{k}^2 3^{k-1}} = \frac{5\pi}{\sqrt{3}} + 6\log 3$ and $\sum_{k=2}^{\infty} \frac{(5k-2)T_{k-1}}{(k-1)k^2 \binom{2k}{k}^2 3^{k-1}} = \frac{21 - 2\sqrt{3}\pi - 9\log 3}{12}$.
• Conjecture 10.2 provides $p$-adic congruences for partial sums of these series modulo $p^2$, such as $p^2 \sum_{k=1}^{p-1} \frac{(105k-44)T_{k-1}}{k^2 \binom{2k}{k}^2 3^{k-1}} \equiv 11\left(\frac{p}{3}\right) + \frac{p}{2}(13 - 35\left(\frac{p}{3}\right)) \pmod{p^2}$.
• The paper conjectures that $\sum_{k=0}^{p-1} \frac{8k+3}{(-16)^k} \binom{2k}{k}^2 T_k(3,-4) \equiv p(1 + 2\left(\frac{-1}{p}\right)) \pmod{p^2}$, linking to new $p$-adic patterns.