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John M. Campbell

Publications and source records attributed to John M. Campbell.

At least 55 records · Page 3Linked to original sources
arXiv2025

Schur-hooks and Bernoulli number recurrences

Given an identity relating families of Schur and power sum symmetric functions, this may be thought of as encoding representation-theoretic properties according to how the $p$-to-$s$ transition matrices provide the irreducible character tables for symmetric groups. The case of the Murnaghan-Nakayama rule for cycles provides that $p_{n} = \sum_{i = 0}^{n-1} (-1)^i s_{(n-i, 1^{i})}$, and, since the power sum generator $p_{n}$ reduces to $ζ(2n)$ for the Riemann zeta function $ζ$ and for specialized values of the indeterminates involved in the inverse limit construction of the algebra of symmetric functions, this motivates both combinatorial and number-theoretic applications related to the given case of the Murnaghan-Nakayama rule. In this direction, since every Schur-hook admits an expansion in terms of twofold products of elementary and complete homogeneous generators, we exploit this property for the same specialization that allows us to express $p_{n}$ with the Bernoulli number $B_{2n}$, using remarkable results due to Hoffman on multiple harmonic series. This motivates our bijective approach, through the use of sign-reversing involutions, toward the determination of identities that relate Schur-hooks and power sum symmetric functions and that we apply to obtain a new recurrence for Bernoulli numbers.

math.CO↗
arXiv2024

On the minimal polynomials of the arguments of dilogarithm ladders

Letting $L_{n}(N, u)$ denote a polylogarithm ladder of weight $n$ and index $N$ with $u$ as an algebraic number, there is a rich history surrounding how mathematical objects of this form can be constructed for a given weight or index. This raises questions as to what minimal polynomials for $u$ are permissible in such constructions. Classical relations for the dilogarithm $\text{Li}_{2}$ provide families of weight-2 ladders in such a way so that the base equations for $u$ consist of a fixed number of terms, and subsequent constructions for dilogarithm ladders rely on sporadic cases whereby $u$ is defined via a cyclotomic equation, as in the supernumary ladders due to Abouzahra and Lewin. This motivates our construction of an infinite family of dilogarithm ladders so as to obtain arbitrarily many terms with nonzero coefficients for the minimal polynomials for $u$. Our construction relies on a derivation of a dilogarithm identity introduced by Khoi in 2014 via the Seifert volumes of manifolds obtained from the use of Dehn surgery.

math.NT↗
arXiv2024

Analytic amplitudes for a pair of Higgs bosons in association with three partons

The pair production of Higgs bosons at the LHC can give information about the triple Higgs boson coupling. We perform an analytic one-loop calculation of the amplitudes for a pair of Higgs bosons in association with three partons, retaining the exact dependence on the quark mass circulating in the loop. These amplitudes constitute the real radiation corrections in the calculation of Higgs boson pair production at next-to-leading order in the strong coupling. The results of an analytic generalised-unitarity computation are simplified via analytic reconstruction in spinor variables. Compact ansätze for kinematic pole residues are iteratively fitted via $p$-adic evaluations near said poles and subtracted until no pole remains. A new ansatz construction is introduced to minimally parametrise coefficients of amplitudes with multiple massive external legs. The simplified expressions are faster to evaluate than automatic codes and can lead to more stable results near singular regions.

hep-ph↗
arXiv2024

A lift of chromatic symmetric functions to $\textsf{NSym}$

If we consider previously introduced extensions of Stanley's chromatic symmetric function $X_{G}(x_1, x_2, \ldots)$ for a graph $G$ to elements in the algebra $\textsf{QSym}$ of quasisymmetric functions and in the algebra $\textsf{NCSym}$ of symmetric functions in noncommuting variables, this motivates our introduction of a lifting of $X_{G}$ to the dual of $\textsf{QSym}$, i.e., the algebra $\textsf{NSym}$ of noncommutative symmetric functions, as opposed to $\textsf{NCSym}$. For an unlabelled directed graph $D$, our extension of chromatic symmetric functions provides an element $\text{X}_{D}$ in $\textsf{NSym}$, in contrast to the analogue $Y_{G} \in \textsf{NCSym}$ of $X_{G}$ due to Gebhard and Sagan. Letting $G$ denote the undirected graph underlying $D$, our construction is such that the commutative image of $\text{X}_{D}$ is $ X_{G}$. This projection property is achieved by lifting Stanley's power sum expansion for chromatic symmetric functions, with the use of the $Ψ$-basis of $\textsf{NSym}$, so that the orderings of the entries of the indexing compositions are determined by the directed edges of $D$. We then construct generating sets for $\textsf{NSym}$ consisting of expressions of the form $\text{X}_{D}$, building on the work of Cho and van Willigenburg on chromatic generating sets for $\textsf{Sym}$.

math.CO↗
arXiv2024

Algorithms for numerically stable scattering amplitudes

The numerically stable evaluation of scattering matrix elements near the infrared limit of gauge theories is of great importance for the success of collider physics experiments. We present a novel algorithm that utilizes double precision arithmetic and reaches higher precision than a naive quadruple precision implementation at smaller computational cost. The method is based on physics-driven modifications to propagators, vertices and external polarizations.

hep-ph↗
arXiv2024

A binary version of the Mahler-Popken complexity function

The (Mahler-Popken) complexity $\| n \|$ of a natural number $n$ is the smallest number of ones that can be used via combinations of multiplication and addition to express $n$, with parentheses arranged in such a way so as to form legal nestings. We generalize $\| \cdot \|$ by defining $\| n \|_{m}$ as the smallest number of possibly repeated selections from $\{ 1, 2, \ldots, m \}$ (counting repetitions), for fixed $m \in \mathbb{N}$, that can be used to express $n$ with the same operational and bracket symbols as before. There is a close relationship, as we explore, between $\|\cdot\|_{2}$ and lengths of shortest addition chains for a given natural number. This illustrates how remarkable it is that $(\| n \|_{2} : n \in \mathbb{N} )$ is not currently included in the On-Line Encyclopedia of Integer Sequences and has, apparently, not been studied previously. This, in turn, motivates our exploration of the complexity function $\| \cdot\|_{2}$, in which we prove explicit upper and lower bounds for $\|\cdot\|_{2}$ and describe some problems and further areas of research concerning $\|\cdot\|_{2}$.

math.NT↗
arXiv2024

A further $q$-analogue of a formula due to Guillera

Hou, Krattenthaler, and Sun have introduced two $q$-analogues of a remarkable series for $π^2$ due to Guillera, and these $q$-identities were, respectively, proved with the use of a $q$-analogue of a Wilf-Zeilberger pair provided by Guillera and with the use of ${}_{3}ϕ_{2}$-transforms. We prove a $q$-analogue of Guillera's formula for $π^2$ that is inequivalent to previously known $q$-analogues of the same formula due to Guillera, including the Hou-Krattenthaler-Sun $q$-identities and a subsequent $q$-identity due to Wei. In contrast to previously known $q$-analogues of Guillera's formula, our new $q$-analogue involves another free parameter apart from the $q$-parameter. Our derivation of this new result relies on the $q$-analogue of Zeilberger's algorithm.

math.CO↗
arXiv2024

A Schur-Weyl duality analogue based on a commutative bilinear operation

Schur-Weyl duality concerns the actions of $\text{GL}_{n}(\mathbb{C})$ and $S_{k}$ on tensor powers of the form $V^{\otimes k}$ for an $n$-dimensional vector space $V$. There are rich histories within representation theory, combinatorics, and statistical mechanics involving the study and use of diagram algebras, which arise through the restriction of the action of $\text{GL}_{n}(\mathbb{C})$ to subgroups of $\text{GL}_{n}(\mathbb{C})$. This leads us to consider further variants of Schur-Weyl duality, with the use of variants of the tensor space $V^{\otimes k}$. Instead of taking repeated tensor products of $V$, we make use of a freest commutative bilinear operation in place of $\otimes$, and this is motivated by an associated invariance property given by the action of $S_{k}$. By then taking the centralizer algebra with respect to the action of the group of permutation matrices in $\text{GL}_{n}(\mathbb{C})$, this gives rise to a diagram-like algebra spanned by a new class of combinatorial objects. We construct orbit-type bases for the centralizer algebras introduced in this paper, and we apply these bases to prove a combinatorial formula for the dimensions of our centralizer algebras.

math.RT↗
arXiv2024

Hypergeometric accelerations with shifted indices

Chu and Zhang, in 2014, introduced hypergeometric transforms derived through the application of an Abel-type summation lemma to Dougall's ${}_{5}H_{5}$-series. These transforms were applied by Chu and Zhang to obtain accelerated rates of convergence, yielding rational series related to the work of Ramanujan and Guillera. We apply a variant of an acceleration method due to Wilf using what we refer to as shifted indices for Pochhammer symbols involved in our first-order, inhomogeneous recurrences derived via Zeilberger's algorithm, to build upon Chu and Zhang's accelerations, recovering many of their accelerated series and introducing many inequivalent series for universal constants, including series of Ramanujan type involving linear polynomials as summand factors, as in Ramanujan's series for $\frac{1}π$.

math.CA↗
arXiv2024

On two-term hypergeometric recursions with free lower parameters

Let $F(n,k)$ be a hypergeometric function that may be expressed so that $n$ appears within initial arguments of inverted Pochhammer symbols, as in factors of the form $\frac{1}{(n)_{k}}$. Only in exceptional cases is $F(n, k)$ such that Zeilberger's algorithm produces a two-term recursion for $\sum_{k = 0}^{\infty} F(n, k)$ obtained via the telescoping of the right-hand side of a difference equation of the form $p_{1}(n) F(n + r, k) + p_{2}(n) F(n, k) = G(n, k+1) - G(n, k)$ for fixed $r \in \mathbb{N}$ and polynomials $p_{1}$ and $p_{2}$. Building on the work of Wilf, we apply a series acceleration technique based on two-term hypergeometric recursions derived via Zeilberger's algorithm. Fast converging series previously given by Ramanujan, Guillera, Chu and Zhang, Chu, Lupaş, and Amdeberhan are special cases of hypergeometric transforms introduced in our article.

math.CA↗
arXiv2024

An extension of the Chudnovsky algorithm

Using an infinite family of generalizations of the Chudnovsky brothers' series recently obtained via the analytic continuation of the Borwein brothers' formula for Ramanujan-type series of level 1, we apply the Gauss-Salamin-Brent iteration for $π$ to obtain a new, Ramanujan-type series that yields more digits per term relative to current world record given by an extension of the Chudnovsky algorithm from Bagis and Glasser that produces about 110 digits per term. We explicitly evaluate the required nested radicals over $\mathbb{Q}$ involved in the our summation, which yields about 153 digits per term, and we provide a practical way of implementing our higher-order version of the Chudnovsky algorithm via the the PARI/GP system. An evaluation due to Berndt and Chan for the modular $j$-invariant associated with their order-3315 extension of the Chudnovskys' Ramanujan-type series provides a key to our applications of recursions for the elliptic lambda and elliptic alpha functions.

math.NT↗
arXiv2024

Multiple elliptic integrals and differential equations

We introduce and prove evaluations for families of multiple elliptic integrals by solving special types of ordinary and partial differential equations. As an application, we obtain new expressions of Ramanujan-type series of level 4 and associated singular values for the complete elliptic integral $\mathbf K$ with integrals involving $\mathbf K$.

math.CA↗
arXiv2024

On a Ramanujan-type series associated with the Heegner number 163

Using the Wolfram NumberTheory package and the Recognize command, together with numerical estimates involving the elliptic lambda and elliptic alpha functions, Bagis and Glasser, in 2013, introduced a conjectural Ramanujan-type series related to the class number $h(-d) = 1$ for a quadratic form with discriminant $d = 163$. This conjectured series is of level one and has positive terms, and recalls the Chudnovsky brothers' alternating series of the same level, given the connection between the Chudnovsky-Chudnovsky formula and the Heegner number $d = 163$ such that $\mathbb{Q}\left( \sqrt{-d} \right)$ has class number one. We prove Bagis and Glasser's conjecture by proving evaluations for $λ^{\ast}(163)$ and $α(163)$, which we derive using the Chudnovsky brothers' formula together with the analytic continuation of a formula due to the Borwein brothers for Ramanujan-type series of level one. As a byproduct of our method, we obtain an infinite family of Ramanujan-type series for $\frac{1}π$ generalizing the Chudnovsky algorithm.

math.NT↗
arXiv2024

Applications of the icosahedral equation for the Rogers-Ramanujan continued fraction

Let $R(q)$ denote the Rogers-Ramanujan continued fraction for $|q| < 1$. By applying the RootApproximant command in the Wolfram language to expressions involving the theta function $f(-q) := (q;q)_{\infty}$ given in modular relations due to Yi, this provides a systematic way of obtaining experimentally discovered evaluations for $R\big(e^{-π\sqrt{r}}\big)$, for $r \in \mathbb{Q}_{> 0}$. We succeed in applying this approach to obtain explicit closed forms, in terms of radicals over $\mathbb{Q}$, for the Rogers-Ramanujan continued fraction that have not previously been discovered or proved. We prove our closed forms using the icosahedral equation for $R$ together with closed forms for and modular relations associated with Ramanujan's $G$- and $g$-functions. An especially remarkable closed form that we introduce and prove is for $R\big( e^{-π\sqrt{48/5} } \big)$, in view of the computational difficulties surrounding the application of an order-25 modular relation in the evaluation of $G_{48/5}$.

math.NT↗
arXiv2023

Proofs of conjectures on Ramanujan-type series of level 3

A Ramanujan-type series satisfies $$ \frac{1}π = \sum_{n=0}^{\infty} \frac{\left( \frac{1}{2} \right)_{n} \left( \frac{1}{s} \right)_{n} \left(1 - \frac{1}{s} \right)_{n} }{ \left( 1 \right)_{n}^{3} } z^{n} (a + b n), $$ where $s \in \{ 2, 3, 4, 6 \}$, and where $a$, $b$, and $z$ are real algebraic numbers. The level $3$ case whereby $s = 3$ has been considered as the most mysterious and the most challenging, out of all possible values for $s$, and this motivates the development of new techniques for constructing Ramanujan-type series of level $3$. Chan and Liaw introduced an alternating analogue of the Borwein brothers' identity for Ramanujan-type series of level $3$; subsequently, Chan, Liaw, and Tian formulated another proof of the Chan-Liaw identity, via the use of Ramanujan's class invariant. Using the elliptic lambda function and the elliptic alpha function, we prove, using a limiting case of the Kummer-Goursat transformation, a new identity for evaluating $z$, $a$, and $b$ for Ramanujan-type series such that $s = 3$ and $z < 0$, and we apply this new identity to prove three conjectured formulas for quadratic-irrational, Ramanujan-type series that had been discovered via numerical experiments with Maple in 2012 by Aldawoud. We also apply our identity to prove a new Ramanujan-type series of level $3$ with quartic values for $z < 0$, $a$, and $b$.

math.NT↗
arXiv2023

A generalization of immanants based on partition algebra characters

We introduce a generalization of immanants of matrices, using partition algebra characters in place of symmetric group characters. We prove that our immanant-like function on square matrices, which we refer to as the recombinant, agrees with the usual definition for immanants for the special case whereby the vacillating tableaux associated with the irreducible characters correspond, according to the Bratteli diagram for partition algebra representations, to the integer partition shapes for symmetric group characters. In contrast to previously studied variants and generalizations of immanants, as in Temperley-Lieb immanants and $f$-immanants, the sum that we use to define recombinants is indexed by a full set of partition diagrams, as opposed to permutations.

math.CO↗
arXiv2023

The prime-counting Copeland-Erdős constant

Let $(a(n) : n \in \mathbb{N})$ denote a sequence of nonnegative integers. Let $0.a(1)a(2)...$ denote the real number obtained by concatenating the digit expansions, in a fixed base, of consecutive entries of $(a(n) : n \in \mathbb{N})$. Research on digit expansions of this form has mainly to do with the normality of $0.a(1)a(2)...$ for a given base. Famously, the Copeland-Erdős constant $0.2357111317...$, for the case whereby $a(n)$ equals the $n^{\text{th}}$ prime number $p_{n}$, is normal in base 10. However, it seems that the ``inverse'' construction given by concatenating the decimal digits of $(π(n) : n \in \mathbb{N})$, where $π$ denotes the prime-counting function, has not previously been considered. Exploring the distribution of sequences of digits in this new constant $0.0122...9101011...$ would be comparatively difficult, since the number of times a fixed $m \in \mathbb{N}$ appears in $(π(n) : n \in \mathbb{N})$ is equal to the prime gap $g_{m} = p_{m+1} - p_{m}$, with the behaviour of prime gaps notoriously elusive. Using a combinatorial method due to Szüsz and Volkmann, we prove that Cramér's conjecture on prime gaps implies the normality of $0.a(1)a(2)...$ in a given base $g \geq 2$, for $a(n) = π(n)$.

math.NT↗