arXiv subjects

Cetin Hakimoglu-Brown

Publications and source records attributed to Cetin Hakimoglu-Brown.

10 recordsLinked to original sources

arXiv2026

Two-term dilogarithms, mixed-base Nahm sums, and Fricke symmetry

We study the arithmetic constraints that two-term Rogers dilogarithm relations impose on rank-two Nahm systems with mixed denominator steps. Monomial complement equations give a uniform conversion to quadratic data in every algebraic degree. Within a specified ten-record simplest-cubic family, positive coupling, positive definiteness, and integer-valued exponents leave, for m > 2, exactly the steps (1,3), (1,13), and (1,31). At the smallest compatible modulus 31, the required support cannot be a multiplicative subgroup coset. Quadratic examples include known product identities and two systems for which the first radial correction excludes every individual modular sum with rational linear terms. For the index-13 system, we prove a two-term Rogers identity and recover its arguments from the matrix of a previously proved two-dimensional Fricke transformation. Its cusp growth agrees with the resulting saddle action, and an independent calculation reproduces all three leading Fricke amplitudes exactly. The associated two-summand product evaluations remain conjectural; the small rank and dimension contrast with the classical rank-five, six-component Andrews-Gordon construction at the same modulus. A mixed-signature cubic family illustrates the limits of the arithmetic restriction

math.NT↗

arXiv2026

A Fricke transformation for cubic-residue eta products of level 13

We prove an explicit Fricke transformation for the two-dimensional space generated by normalized reciprocal products on the cubic-residue cosets modulo 13. The transformation matrix is a scalar multiple of a matrix of differences of cubic Gaussian periods. Its projective action is defined over the cyclic cubic field, whereas the normalized matrix is defined over the real cyclotomic field. The proof uses generalized eta functions, a modular unit of degree two on $X_1(13)$, and a five-coefficient identity. A separate divisor argument lifts the projective transformation to the asserted linear transformation. We also record the Fricke images at an arbitrary prime $p\equiv1\pmod3$ and prove that their normalized sine constants belong to the associated cyclic cubic subfield.

math.NT↗

arXiv2026

On a conjecture due to Kanade related to Nahm sums

Kanade explored the construction of modular companions to $q$-series identities, using the asymptotics of Nahm sums, and Mizuno [Ramanujan J. 66 (2025), Article 62, 31 pp.] recently obtained a generalization of Kanade's asymptotic formula for symmetrizable Nahm sums. A related conjecture from Kanade concerning the dilogarithm function and related to the work of Kurşungöz on Andrews--Gordon-type series [Ann. Comb. 23 (2019), 835--888] has remained open. In this paper, we prove Kanade's conjecture, through an application of dilogarithm identities due to Kirillov together with a dilogarithm ladder due to Loxton and Lewin. Inspired by Kanade's result, we extend this to prove two new related dilogarithm identities.

math.NT↗

arXiv2026

Block and Residue Exclusion in Kernel-Root Expansions for Two-Barrier Walks

We consider the two-barrier walk on the integers with steps $+y$ and $-b$, stopped on entering either absorbing set. Its finite-state generating functions are rational and admit the standard Schur-function representation in the roots of the normalized kernel $P_z(X)=X^{b+y}-2X^b+z$. Our contribution is a two-stage pruning method for constructing these root expansions without computing terms that cannot affect the final polynomial. First, the Weyl expansion is grouped into complete small-root/large-root blocks, and a $z$-adic valuation bound removes every block beginning above the denominator degree. Second, surviving blocks are split into residue subblocks and projected onto complete cyclic root-label orbits; the resulting congruences eliminate nontrivial characters without any termwise rationality test. In the $(+3,-2)$, $m=12$ example only one mixed-block degree survives, while for $(+5,-2)$, $m=22$, twenty of twenty-one Weyl terms are excluded before expansion.

math.PR↗

arXiv2026

Collapsing Fifty Dilogarithm Arguments to Five Terms over $\mathbb{Q}\bigl(u,\sqrt{4-3u^{2}}\bigr)$

We give a one-parameter functional equation for the real Rogers dilogarithm, with arguments in $\mathbb{Q}\bigl(u,\sqrt{4-3u^{2}}\bigr)$, and prove it by an explicit array of ten instances of Rogers' five-term relation: the fifty arguments so contributed cancel down to the five of the identity, which admit no shorter relation among themselves. The equation comes from a pair of integrals whose equality is elementary, and we show that the underlying integrand is essentially forced. Specialising the parameter gives identities over $\mathbb{Q}(\sqrt{33})$ and $\mathbb{Q}(\sqrt{17})$, a relation between $\mathbb{Q}(\sqrt{13})$, $\mathbb{Q}(\sqrt{3})$, and $\operatorname{Cl}_2(π/3)$ with a new analogue for Catalan's constant, and a pair of dilogarithm ladders of quartic base over $\mathbb{Q}(\sqrt{33})$. The base equations of these ladders, and of four conjectural ones located by integer-relation search, are irreducible quartics with four real roots. All previously recorded ladders of degree four known to us have base equations with two real roots, so these appear to be the first totally real ladders of degree exceeding three.

math.NT↗

arXiv2026

${}_5F_4$ evaluations and a family of $π^2+\log^2$ identities

We evaluate the series $\sum_{n\ge1} z^n\big/\!\big(n^2\binom{4n}{n}\big)$, equal to $-\tfrac{z}{4}\,{}_5F_4\!\left(1,1,1,\tfrac43,\tfrac53;\tfrac54,\tfrac32,\tfrac74,2;\tfrac{27z}{256}\right)$, in closed form at an infinite family of algebraic points indexed by a rational angle $θ=jπ/N$. Each value equals $c\,π^2$ plus a universal rational quadratic form in three logarithms, with $c=-\tfrac13\left(1-\tfrac{2j}{N}\right)^2$. This is the quartic-base case reached but not evaluated by D'Aurizio and Di Trani. The proof is self-contained: an exact integer factor relating two weights, followed by Landen's identity, reduces the integral to a sum of squared logarithms.

math.NT↗

arXiv2026

Functional Dilogarithm Identities in Quadratic Fields

We derive three- and six-term functional dilogarithm identities whose arguments lie in $\mathbb{Q}(u,\sqrt{4-3u^2})$ and $\mathbb{Q}(u,\sqrt{u(4-3u)})$. Our approach is based on an integral-to-${}_4F_3$ correspondence that converts families of cubic and sextic integrals into hypergeometric identities, providing a systematic method for constructing functional equations for the dilogarithm over quadratic fields. We demonstrate the power of this method by giving an analytic proof of the classical Loxton--Lewin $2\cos(4π/9)$ identity, deriving a new family of dilogarithm ladders of quartic base lying in $\mathbb{Q}(\sqrt{33})$, and proving conjectural two-term identities of Bytsko. As a further application, we obtain rapidly convergent ${}_4F_3$ series for $\mathrm{Cl}_2(π/3)$ and explicit relations connecting $\mathbb{Q}(\sqrt{13})$ and $\mathbb{Q}(\sqrt{3})$. Finally, a PSLQ-based search over palindromic quartic units yields new ladder relations with arguments built from $2\tan(π/8)\cos(π/5)$ and $\tan(3π/20)$, analogous to known trigonometric identities of Watson, Loxton, and Gordon--McIntosh.

math.CA↗

arXiv2026

On a $_2F_1\big(\frac{1}{4}\big)$-identity due to Gosper

It is only in exceptional cases that a $_2F_1(z)$-series with rational parameters and a rational argument, apart from the cases for $z \in \{ \pm 1, \frac{1}{2} \}$ associated with classical hypergeometric identities, admits an evaluation given by a combination of $Γ$-values with rational arguments. In this paper, we present a new and integration-based approach toward the construction of special values for $_2F_1$-series of the desired form. We apply this approach using a $_2F_1\big(\frac{1}{4}\big)$-identity originally due to Gosper and later considered by Vidunas, Ebisu, and Zudilin, to evaluate a ${}_{2}F_{1}$-series of convergence rate $\big(\frac{172872}{185039}\big)^2$. With regard to extant research on so-called ``strange'' ${}_{2}F_{1}$-evaluations, as in the work of Ebisu and Zeilberger, our new series seems to have the largest numerator/denominator in its argument.

math.CA↗

arXiv2025

Analytic Dilogarithm Identities

We introduce dilogarithm identities through a beta integral-based technique that we apply to provide analytic proofs of previously conjectured dilogarithm relations, solving open problems given by both Bytsko and Campbell, and that we further apply to construct and prove new ladder relations with quartic and sextic bases. We also apply our method to introduce and prove two-term $\operatorname{Li}_2$-relations and ladder-like identities with arguments in algebraic number fields such as $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{3})$, $\mathbb{Q}(\sqrt{5})$, and $\mathbb{Q}(\sqrt{-7})$. Moreover, single-term dilogarithm evaluations are introduced and derived.

math.NT↗

arXiv2023

Generating Functions for Restricted Motzkin Paths in a Slit for Arbitrary Weights

In this paper, I derive a generating function for discrete Motzkin paths of step size +2,+1,-1, restricted between two absorbing parallel lines, and generalized to arbitrary step weights. This continues off work of similar problems involving directed paths with either single barriers, uniform step size, or uniform weights.

math.CO↗