Various conjectural series identities

Abstract: In this paper we collect over 100 new series identities (involving binomial coefficients) conjectured by the author in 2026. Most of them involve $π$ or Riemann's zeta function. For example, we conjecture that $$\sum_{k=0}^\infty\frac{16k+3}{(-202^2)^k}\binom{2k}kT_k(19,-20)T_{2k}(9,-5)=\frac{43\sqrt{101}}{75π},$$ where $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. The conjectures in this paper might interest some readers and stimulate further research.

• The paper compiles over 100 conjectural series identities that generalize Ramanujan-type formulas using binomial coefficients, generalized trinomial coefficients, and harmonic numbers.
• The paper introduces new families of rapidly converging 1/π series and harmonic number identities, supported by high-precision computations and companion p-adic congruences.
• The paper reveals deep connections between analytic series evaluations and arithmetic structures, linking the results to modular forms, L-functions, and quadratic field representations.

Summary of “Various conjectural series identities” (2603.29973)

Introduction and Scope

The paper by Zhi-Wei Sun compiles over 100 conjectural series identities discovered in 2026, primarily involving binomial coefficients, generalized central trinomial coefficients, harmonic numbers, Riemann zeta values, and related constants such as Catalan’s constant and Dirichlet L-functions. Many conjectures connect Ramanujan-style rapidly converging series to explicit evaluations involving $\pi$, zeta values, and logs, and include congruence relations modulo prime powers.

The identities are presented without proof, and are supported by extensive high-precision computations. The paper delineates several new classes of series, introduces new types of summands, and provides wide-reaching generalizations of previously established Ramanujan-type formulas. It also presents a strong interplay between analytic conjectural evaluations and arithmetic congruences.

New Ramanujan-type Series for $1/\pi$ Involving Generalized Central Trinomial Coefficients

A central focus is the conjecture of a new “Type X” family of $1/\pi$ series, each of the form

$$\sum_{k=0}^\infty \frac{a + d k}{m^k} \binom{2k}{k} T_k(b,c) T_{2k}(b_*, c_*),$$

where $T_n(b,c)$ is the $n$-th generalized central trinomial coefficient (the coefficient of $x^n$ in $(x^2 + b x + c)^n$). These generalize both the classical binomial sums and the types introduced in the author’s prior work.

The paper explicitly formulates three new conjectural formulas with striking evaluations, such as:

$$\sum_{k=0}^\infty \frac{16k+3}{(-202^2)^k} \binom{2k}{k} T_k(19,-20) T_{2k}(9,-5) = \frac{43\sqrt{101}}{75\pi}$$

and analogous identities with different polynomial arguments and parameters.

Furthermore, for each analytic identity, Sun proposes companion $p$-adic congruences for truncated sums, involving intricate expressions modulo $1/\pi$0 and links to quadratic fields via representations of primes.

Conjectural Series Involving Harmonic Numbers

Sun extends the spectrum of Ramanujan-type identities to include sums with harmonic numbers and their generalizations (e.g., $1/\pi$1), reflecting an ongoing trend in the interplay between hypergeometric sums and multiple zeta values.

For instance, identities of the form

$1/\pi$2

are conjectured, generalizing known Ramanujan series by inserting nontrivial combinations of harmonic numbers in the summand. Numerous variants for different moduli and argument shuffles are posed.

This theme is carried further into identities yielding closed forms for $1/\pi$3 type terms, with right sides referencing $1/\pi$4, $1/\pi$5, and their products.

Series Involving Higher-order Derivatives and Gamma Functions

The work advances to series whose summands involve higher derivatives of functions of the form

$1/\pi$6

evaluated at positive integers, paralleling recent developments in using the Wilf-Zeilberger (WZ) method and generating relations among values of zeta-type functions.

For example,

$1/\pi$7

where $1/\pi$8 is a Dirichlet $1/\pi$9-function. Further, higher derivatives relate to explicit combinations of zeta values, multiple zeta values $1/\pi$0, and their algebraic multiples.

This systematic compilation of conjectural evaluations embraces both alternating and non-alternating series, and in some cases, the imaginary parts of sums involving complex exponentials, highlighting the reach of these heuristic discoveries.

Congruences and Arithmetical Structure

A notable aspect of the paper is the systematic pairing of analytic identities with supercongruence conjectures involving truncated sums modulo prime powers, in the style of Rodriguez-Villegas and Tauraso. The congruences often depend on representations of primes by binary quadratic forms, nontrivially linking the analytic formulas to deep arithmetic structure.

Claims include that the sums (modulo $1/\pi$1) correspond to values dependent on Legendre symbols, with parameters reflecting the splitting of $1/\pi$2 in certain quadratic fields, indicating a strong modular and motivic underpinning.

Implications and Future Directions

The compiled conjectures considerably broaden the known experimental landscape of Ramanujan-type identities, incorporating series with trinomial coefficients, multiple polynomial arguments, harmonic numbers, and high-order derivatives. The conjectures point toward new structural phenomena and potential deeper connections with modular forms, $1/\pi$3-functions, and WZ-theory.

Practically, the fast convergence of these series suggest applications in high-precision computation of $1/\pi$4, zeta values, and related constants provided the analytic conjectures can be resolved rigorously. The arithmetic congruences rendered with each analytic identity may provide test beds for $1/\pi$5-adic, motivic, or automorphic generalizations.

The main theoretical implication is the suggestion that the algebraic and modular anatomy underlying Ramanujan-type identities is even richer than previously imagined, with explicit links to derivatives, multiple zeta values, and $1/\pi$6-adic properties. The symbiosis between analytic and arithmetic phenomena evident in these conjectures is likely to stimulate further investigation, both computationally and theoretically, in symbolic computation, number theory, and experimental mathematics.

Future progress may come from a combination of algorithmic approaches (such as the extended WZ method for multiple sums and derivatives), modular parametrizations, deeper studies of special function identities, or perhaps new motivic interpretations.

Conclusion

The paper presents an extensive trove of conjectural series identities, blending analytic evaluation with arithmetic structure and offering novel directions for research at the intersection of experimental mathematics, combinatorics, and analytic number theory. By systematically exploring and conjecturing formulas for an expanded class of summands and parameters, the work both generalizes and unifies themes in Ramanujan-type series, supercongruences, and $1/\pi$7-function values, mapping out fertile territory for future breakthroughs in the explicit evaluation of special values and series.