arXiv subjects

John M. Campbell

Publications and source records attributed to John M. Campbell.

At least 37 records · Page 2Linked to original sources

arXiv2026

Kronecker coefficients via the Giambelli identity for Schur functions

One of the central open problems in both algebraic combinatorics and representation theory is to find a positive combinatorial rule for Kronecker coefficients $ g_{λ\, μ\, ν}$. A notable advance in this direction is due to Blasiak, who proved a combinatorial interpretation in terms of colored Yamanouchi tableaux for the case whereby one of the indexing partitions is hook-shaped. In this paper, we introduce a framework for the evaluation and combinatorial interpretation of Kronecker coefficients, combining a Schur function identity of Littlewood, the Giambelli identity for Schur functions, and Blasiak's combinatorial rule. This framework reduces the study of Kronecker coefficients to alternating sums involving hook-indexed cases. As an application of this framework, we obtain combinatorial interpretations of $g_{t, h^{(1)}, h^{(2)}}$ for two-row partitions $t$ and hook-like partitions $h^{(1)}$ and $h^{(2)}$ satisfying natural conditions. More broadly, our approach provides a systematic method for extending hook-based combinatorial rules to wider families of Kronecker coefficients.

math.CO↗

arXiv2026

A further q-analogue of Gosper's strange series

Recently, the second author [Ramanujan J. 2026] introduced and proved a $q$-series identity that appears to provide the first known $q$-analogue of an evaluation for a ${}_{2}F_{1}$-series known as \emph{Gosper's strange series}. Yamaguchi's derivation of this $q$-analogue relies on three-term relations for ${}_{2}ϕ_{1}$-series along with Heine's transformation of ${}_{2}ϕ_{1}$-series. In this note, we introduce and prove, using a $q$-analogue of a series evaluation technique relying on an Abel-type summation lemma, a further $q$-analogue of Gosper's ${}_{2}F_{1}$-identity that is inequivalent to Yamaguchi's $q$-analogue, and we also apply this technique to construct an alternative and simplified proof of Yamaguchi's $q$-analogue, together with a ${}_{3}ϕ_{2}$-series variant of Heine's $q$-analogue of Gauss's hypergeometric formula, a ${}_{6}ϕ_{5}$-series variant and two ${}_{4}ϕ_{3}$-series variants of the $q$-analogue of Kummer's identity due to Bailey and Daum, along with a $q$-analogue of a result obtained by Cantarini [Ramanujan J. 2022] via Fourier-Legendre theory and related to Ramanujan's first series for $\frac{1}π$.

math.CA↗

arXiv2026

The construction of $q$-analogues via $_3ϕ_2$-series and $q$-difference equations

We apply the EKHAD-normalization method given in our recent work to obtain, via the $q$-version of Zeilberger's algorithm, $q$-WZ pairs $(F, G)$ such that $\sum_{k = 0}^{\infty} F(0, k)$ may be expressed as a basic hypergeometric series of the form ${}_{3}ϕ_2$ with multiple free parameters, and in such a way so that $\sum_{k=0}^{\infty} F(0, k) = \sum_{n=0}^{\infty} G(n, 0)$. In contrast to how previous applications of EKHAD-normalization relied on $q$-analogues for specific WZ pairs introduced by Guillera, our multiparameter approach provides a broad framework in the construction of $q$-analogues for accelerated series for universal constants such as $π$. We apply this multiparameter version of EKHAD-normalization to obtain and prove new $q$-analogues for accelerated hypergeometric series attributed to many authors, including (alphabetically) Adamchik and Wagon, Apéry, Chu, Chu and Zhang, Fabry, Guillera, Ramanujan, and Zeilberger.

math.CO↗

arXiv2026

Quadratic irrational analogues of Ramanujan's series for $1/π$

About 40 years ago Jonathan and Peter Borwein discovered the series identity $$ \sum_{n=0}^\infty \frac{(-1)^n(6n)!}{(3n)!(n!)^3} \frac{(A+nB)}{C^{n+1/2}} = \frac{1}{12π} $$ where \begin{align*} A&=1657145277365+212175710912\sqrt{61},\cr B&=107578229802750+13773980892672\sqrt{61},\cr C&=\left(5280(236674+30303\sqrt{61})\right)^3 \end{align*} which adds roughly 25 digits of accuracy per term. They noted that if each of the quadratic irrationals $A$, $B$ and $C$ is replaced by their conjugates, that is, each number $a+b\sqrt{61}$ is changed to $a-b\sqrt{61}$, then the resulting series also converges to a rational multiple of $1/π$. They gave several other examples of quadratic irrational series for $1/π$, and noted that the conjugate series converges to another rational multiple of $1/π$ or in some cases the conjugate series diverges. The purpose of this work is to provide an explanation and classification of such series. Our classification includes Ramanujan's 17 original series, as well as series of the Borweins, Chudnovskys, Sato and others. We extend the classification to genus-zero subgroups $Γ_0(\ell)+$, that is, for each $\ell \in \big\{1,2,3,\ldots,36,38,39,41,42,44,45,46,47,49,50,51,54,55,56,59,60,62,66,69,70, 71,78,87,92,94,95,105,110,119\big\}$ we calculate the Hauptmoduls, associated weight two modular forms, and the corresponding rational and real quadratic irrational series for $1/π$. The classification reveals many interrelations among the different series. For example, we show that the Borweins' series above, and its conjugate, are equivalent by hypergeometric transformation formulas to the level~7 rational series $$ \sum_{n=0}^\infty \left\{\sum_{j=0}^n {n \choose j}^2{2j \choose n} {n+j \choose j}\right\} (11895n+1286) \frac{(-1)^n}{22^{3n+3}} = \frac{1}{π\sqrt{7}}. $$

math.NT↗

arXiv2026

QCD splitting functions beyond kinematical limits

We present a systematic decomposition of QCD splitting functions into scalar dipole radiators and pure splitting remainders up to second order in the strong coupling. The individual components contain terms that are formally sub-leading in soft or collinear scaling parameters, but well understood and universal due to their origin in scalar QCD. The multipole radiator functions which we derive share essential features of the known double-soft and one-loop soft gluon currents, and are not based on kinematical approximations.

hep-ph↗

arXiv2025

Lexical tableaux and quasisymmetric functions

There is a natural bijection between standard immaculate tableaux of composition shape $α\vDash n$ and length $\ell(α) = k$ and the $ \left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\} $ set-partitions of $\{ 1, 2, \ldots, n \}$ into $k$ blocks, for the Stirling number $ \left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\} $ of the second kind. We introduce a family of tableaux that we refer to as \emph{lexical tableaux} that generalize immaculate tableaux in such a way that there is a bijection between standard lexical tableaux of shape $α\vDash n$ and length $\ell(α) = k$ and the $ \left[ \begin{smallmatrix} n \\ k \end{smallmatrix} \right] $ permutations on $\{ 1, 2, \ldots, n \}$ with $k$ disjoint cycles. In addition to the entries in the first column strictly increasing, the defining characteristic of lexical tableaux is that the word $w$ formed by the consecutive labels in any row is the lexicographically smallest out of all cyclic permutations of $w$. This includes weakly increasing words, and thus, lexical tableaux provide a natural generalization of immaculate tableaux. Extending this generalization, we introduce a pair of dual bases of the Hopf algebras $\textsf{QSym}$ and $\textsf{NSym}$ defined in terms of lexical tableaux. We present two expansions of these bases, involving the monomial and fundamental bases (or, dually, the ribbon and complete homogeneous bases), using Kostka coefficient analogues and coefficients derived from standard lexical tableaux.

math.CO↗

arXiv2025

Reduced complexities for sequences over finite alphabets

Letting $w$ denote a finite, nonempty word, let $\text{red}(w)$ denote the word obtained from $w$ by replacing every subword $s$ of $w$ of the form $cc \cdots c$ for a given character $c$ (such that there is no character immediately to the left or right of $s$ equal to $c$) with $c$. Complexity functions for infinite words play important roles within combinatorics on words, and this leads us to introduce and investigate variants of the factor and abelian complexity functions using the given reduction operation. By enumerating words $v$ and $w$ of a given length $n \geq 0$ and associated with an infinite sequence over a finite alphabet such that $\text{red}(v)$ and $\text{red}(w)$ are equal or otherwise equivalent in some specified way, by analogy with the factor and abelian complexity functions, this may be seen as producing simplified versions of previously introduced complexity functions. We prove a recursion for the reduced factor complexity function $ρ_{\mathbf{t}}^{\text{red}}$ for the Thue-Morse sequence $\mathbf{t}$, giving us that $(ρ_{\mathbf{t}}^{\text{red}}(n) : n \in \mathbb{N})$ is a $2$-regular sequence, we prove an explicit evaluation for the reduced factor complexity function $ρ_{\mathbf{f}}^{\text{red}}$ for the (regular) paperfolding sequence $\mathbf{f}$, together with an evaluation for the reduced abelian complexity function $ρ_{\mathbf{f}}^{\text{ab}, \text{red}}$ for $\mathbf{f}$. We conclude with open problems concerning $ρ_{\mathbf{t}}^{\text{ab}, \text{red}}$.

math.CO↗

arXiv2025

$q$-analogues of $π$-formulas due to Ramanujan and Guillera

The first known $q$-analogues for any of the $17$ formulas for $\frac{1}π$ due to Ramanujan were introduced in 2018 by Guo and Liu (J. Difference Equ. Appl. 29:505-513, 2018), via the $q$-Wilf-Zeilberger method. Through a "normalization" method, which we refer to as EKHAD-normalization, based on the $q$-polynomial coefficients involved in first-order difference equations obtained from the $q$-version of Zeilberger's algorithm, we introduce $q$-WZ pairs that extend WZ pairs introduced by Guillera (Adv. in Appl. Math. 29:599-603, 2002) (Ramanujan J. 11:41-48, 2006). We apply our EKHAD-normalization method to prove four new $q$-analogues for three of Ramanujan's formulas for $\frac{1}π$ along with $q$-analogues of Guillera's first two series for $\frac{1}{π^2}$. Our normalization method does not seem to have been previously considered in any equivalent way in relation to $q$-series, and this is substantiated through our survey on previously known $q$-analogues of Ramanujan-type series for $\frac{1}π$ and of Guillera's series for $\frac{1}{π^2}$. We conclude by showing how our method can be adapted to further extend Guillera's WZ pairs by introducing hypergeometric expansions for $\frac{1}{π^2}$.

math.CA↗

arXiv2025

Analytic reconstruction with massive particles: one-loop amplitudes for $0 \to \bar{q}qt\bar{t}H$

We present an analytic reconstruction of one-loop amplitudes for the process $0 \to \bar{q}qt\bar{t}H$. Our calculation is a novel use of analytic reconstruction, retaining explicit covariance in the massive spin states through the massive spinor-helicity formalism. The analytic reconstruction relies on embedding the massive five-point kinematics in a fully massless eight-point phase space while still building a minimal ansatz directly in the five-point phase space. In order to obtain compact analytic expressions it is necessary to identify suitable partial fraction decompositions and extract common numerator factors, which we achieve through careful inspection of limits in which pairs of denominators vanish. We find that the resulting amplitudes are more numerically efficient than ones computed using automatic methods but that the gains are not as significant as in the massless case, at least at present. The method opens the door to applications at two-loop order, where numerical efficiency and improvements in the reconstruction methodology are more crucial, especially with regards to the number of free parameters in the ansatz.

hep-ph↗

arXiv2025

An analytic result for the $0 \to ggHHH$ amplitude

We present a fully analytic calculation of the leading-order one-loop amplitude for triple Higgs production via gluon fusion, $gg \to HHH$, retaining full dependence on the mass of the heavy quark circulating in the loop. This amplitude provides a direct probe of the triple and quartic Higgs self-couplings, the measurement of which is a central goal of current and future colliders. The amplitude can be presented in compact form thanks to the use of analytic reconstruction techniques, based on finite-field and $p$-adic evaluations, partial fraction decompositions, and primary decompositions to identify common numerator factors. Our results provide a compact and efficient representation of the matrix element for this process, enabling evaluations that are more than an order of magnitude faster than existing numerical alternatives.

hep-ph↗

arXiv2025

Partition algebras as monoid algebras

Wilcox has considered a twisted semigroup algebra structure on the partition algebra $\mathbb{C}A_k(n)$, but it appears that there has not previously been any known basis that gives $\mathbb{C}A_k(n)$ the structure of a "non-twisted" semigroup algebra or a monoid algebra. This motivates the following problem, for the non-degenerate case whereby $n \in \mathbb{C} \setminus \{ 0, 1, \ldots, 2 k - 2 \}$ so that $ \mathbb{C}A_k(n)$ is semisimple. How could a basis $M_{k} = M$ of $ \mathbb{C}A_k(n)$ be constructed so that $M$ is closed under the multiplicative operation on $\mathbb{C}A_k(n)$, in such a way so that $M$ is a monoid under this operation, and how could a product rule for elements in $M$ be defined in an explicit and combinatorial way in terms of partition diagrams? We construct a basis $M$ of the desired form using Halverson and Ram's matrix unit construction for partition algebras, Benkart and Halverson's bijection between vacillating tableaux and set-partition tableaux, an analogue given by Colmenarejo et al. for partition diagrams of the RSK correspondence, and a variant of a result due to Hewitt and Zuckerman characterizing finite-dimensional semisimple algebras that are isomorphic to semigroup algebras.

math.CO↗

arXiv2025

Semisimple algebras related to immaculate tableaux

Given a direct sum $A$ of full matrix algebras, if there is a combinatorial interpretation associated with both the dimension of $A$ and the dimensions of the irreducible $A$-modules, then this can be thought of as providing an analogue of the famous Frobenius-Young identity $n! = \sum_{λ\vdash n} ( f^λ )^{2}$ derived from the semisimple structure of the symmetric group algebra $\mathbb{C}S_{n}$, letting $f^λ$ denote the number of Young tableaux of partition shape $λ\vdash n$. By letting $g^α$ denote the number of standard immaculate tableaux of composition shape $α\vDash n$, we construct an algebra $\mathbb{C}\mathcal{I}_{n}$ with a semisimple structure such that $\dim \mathbb{C}\mathcal{I}_{n} = \sum_{α\vDash n} (g^α)^{2}$ and such that $\mathbb{C}\mathcal{I}_{n} $ contains an isomorphic copy of $\mathbb{C}S_{n}$. We bijectively prove a recurrence for $\dim \mathbb{C}\mathcal{I}_{n}$ so as to construct a basis of $\mathbb{C}\mathcal{I}_{n}$ indexed by permutation-like objects that we refer to as immacutations. We form a basis $\mathcal{B}_{n}$ of $\mathbb{C}\mathcal{I}_{n}$ such that $\mathbb{C} \mathcal{B}_n$ has the structure of a monoid algebra in such a way so that $\mathcal{B}_n$ is closed under the multiplicative operation of $\mathbb{C} \mathcal{I}_n$, yielding a monoid structure on the set of order-$n$ immacutations.

math.CO↗

arXiv2025

A generalization of Deterministic Finite Automata related to discharging

Deterministic Finite Automata (DFAs) are of central importance in automata theory. In view of how state diagrams for DFAs are defined using directed graphs, this leads us to introduce a generalization of DFAs related to a method widely used in graph theory referred to as the discharging method. Given a DFA $(Q, Σ, δ, q_{0}, F)$, the transition function $δ\colon Q \times Σ\to Q$ determines a directed path in the corresponding state diagram based on an input string $a_{1} a_{2} \cdots a_{n}$ consisting of characters in $Σ$, and our generalization can be thought of as being based on how each vertex in $D$ ''discharges'' rational values to adjacent vertices (by analogy with the discharging method) depending on the string $a_{1} a_{2} \cdots a_{n}$ and according to a fixed set of rules. We formalize this notion and pursue an exploration of the notion of a Discharging Deterministic Finite Automaton (DDFA) introduced in this paper. Our DDFA construction gives rise to a ring structure consisting of sequences that we refer to as being quasi-$k$-regular, and this ring generalizes the ring of $k$-regular sequences introduced by Allouche and Shallit.

cs.FL↗

arXiv2025

An iterative approach toward hypergeometric accelerations

Each of Ramanujan's series for $\frac{1}π$ is of the form $$ \sum_{n=0}^{\infty} z^n \frac{ (a_{1})_{n} (a_{2})_{n} (a_{3})_{n} }{ (b_{1})_{n} (b_{2})_{n} (b_{3})_{n} } (c_{1} n + c_2) $$ for rational parameters such that the difference between the arguments of any lower and upper Pochhammer symbols is not an integer. In accordance with the work of Chu, if an infinite sum of this form admits a closed form, then this provides a formula of Ramanujan type. Chu has introduced remarkable results on formulas of Ramanujan type, through the use of accelerations based on $Ω$-sums related to classical hypergeometric identities. Building on our past work on an acceleration method due to Wilf relying on inhomogeneous difference equations derived from Zeilberger's algorithm, we extend this method through what we refer to as an iterative approach that is inspired by Chu's accelerations derived using iteration patterns for well-poised $Ω$-sums and that we apply to introduce and prove many accelerated formulas of Ramanujan type for universal constants, along with many further accelerations related to the discoveries of Ramanujan, Guillera, and Chu.

math.NT↗

arXiv2025

The reflection complexity of sequences over finite alphabets

In combinatorics on words, the well-studied factor complexity function $ρ_{\infw{x}}$ of a sequence $\infw{x}$ over a finite alphabet counts, for every nonnegative integer $n$, the number of distinct length-$n$ factors of $\infw{x}$. In this paper, we introduce the \emph{reflection complexity} function $r_{\infw{x}}$ to enumerate the factors occurring in a sequence $\infw{x}$, up to reversing the order of symbols in a word. We prove a number of results about the growth properties of $r_{\infw{x}}$ and its relationship with other complexity functions. We also prove a Morse--Hedlund-type result characterizing eventually periodic sequences in terms of their reflection complexity, and we deduce a characterization of Sturmian sequences. We investigate the reflection complexity of quasi-Sturmian, episturmian, $(s+1)$-dimensional billiard, complementation-symmetric Rote, and rich sequences. Furthermore, we prove that if $\infw{x}$ is $k$-automatic, then $r_{\infw{x}}$ is computably $k$-regular, and we use the software \texttt{Walnut} to evaluate the reflection complexity of some automatic sequences, such as the Thue--Morse sequence. We note that there are still many unanswered questions about this reflection measure.

math.CO↗

arXiv2025

Extensions of the truncated pentagonal number theorem

Andrews and Merca introduced and proved a $q$-series expansion for the partial sums of the $q$-series in Euler's pentagonal number theorem. Kolitsch, in 2022, introduced a generalization of the Andrews-Merca identity via a finite sum expression for $ \sum_{n \geq k} \frac{ q^{ (k + m) n } }{ \left( q; q \right)_{n} } \left[ \begin{smallmatrix} n - 1 \\ k - 1 \end{smallmatrix} \right]_{q}$ for positive integers $m$, and Yao also proved an equivalent evaluation for this $q$-series in 2022, and Schlosser and Zhou extended this result for complex values $m$ in 2024, with the $m = 1$ case yielding the Andrews-Merca identity, and with the $m = 2$ case having been proved separately by Xia, Yee, and Zhao. We introduce and apply a method, based on the $q$-version of Zeilberger's algorithm, that may be used to obtain finite sum expansions for $q$-series of the form $ \sum_{n \geq 1} \frac{ q^{ p(k) n } }{ \left( q; q \right)_{n + \ell_2} } \left[ \begin{smallmatrix} n - \ell_{1} \\ k - 1 \end{smallmatrix} \right]_{q} $ for linear polynomials $p(k)$ and $\ell_{1} \in \mathbb{N}$ and $\ell_{2} \in \mathbb{N}_{0}$, thereby generalizing the Andrews-Merca identity and the Kolitsch, Yao, and Schlosser-Zhou identities. For example, the $(p(k), \ell_1, \ell_2) = (k+1, 2, 0)$ case provides a new truncation identity for Euler's pentagonal number theorem.

math.NT↗

arXiv2025

Three-parameter generalizations of formulas due to Guillera

Guillera has introduced remarkable series expansions for $\frac{1}{π^2}$ of convergence rates $-\frac{1}{1024}$ and $-\frac{1}{4}$ via the Wilf-Zeilberger method. Through an acceleration method based on Zeilberger's algorithm and related to Chu and Zhang's series accelerations based on Dougall's ${}_{5}H_{5}$-series, we introduce and prove three-parameter generalizations of Guillera's formulas. We apply our method to construct rational, hypergeometric series for $\frac{1}{π^2}$ that are of the same convergence rates as Guillera's series and that have not previously been known.

math.CA↗

arXiv2025

Quasi-immanants

For an integer partition $ λ$ of $n$ and an $n \times n$ matrix $A$, consider the expansion of the immanant $\text{Imm}^λ(A)$ as a sum indexed by permutations $σ$ of order $n$, with coefficients given by the irreducible characters $χ^λ(\text{ctype}(σ))$ of the symmetric group $S_{n}$, for the cycle type $\text{ctype}(σ) \vdash n$ of $σ$. Skandera et al. have introduced combinatorial interpretations of a generalization of immanants given by replacing the coefficient $χ^λ(\text{ctype}(σ))$ with preimages with respect to the Frobenius morphism of elements among the distinguished bases of the algebra $\textsf{Sym}$ of symmetric functions. Since $ \textsf{Sym}$ is contained in the algebra $\textsf{QSym}$ of quasisymmetric functions, this leads us to further generalize immanants with the use of quasisymmetric functions. Since bases of $ \textsf{QSym}$ are indexed by integer compositions, we make use of cycle compositions in place of cycle types to define the family of quasi-immanants introduced in this paper. This is achieved through the use of the quasisymmetric power sum bases due to Ballantine et al., and we prove a combinatorial formula for the coefficients arising in an analogue, given by a special case of quasi-immanants associated with quasisymmetric Schur functions, of second immanants.

math.CO↗