arXiv subjects

Per-Håkan Lundow

Publications and source records attributed to Per-Håkan Lundow.

4 recordsLinked to original sources

arXiv2026

On $(p,q)$-binomial coefficient ratios for complex parameters

We prove local asymptotics for near-central complex $(p,q)$-binomial coefficient moduli ratios allowing an imaginary parameter perturbation of order $n^{-3/4}$ at a $\sqrt{n}$ length scale from the centre. Moreover, we obtain ratio asymptotics for a smaller imaginary perturbation of order $n^{-5/4}$ at the length scale $n^{3/4}$. These results were obtained by reducing the two-parameter coefficients to just one parameter, giving a branch-free logarithmic representation of the second-order ratio and, hence, uniform complex curvature asymptotes for near-central ratios.

math.CA↗

arXiv2026

Quartic Fourier-Laplace limits for $(p,q)$-Rogers-Szegő polynomials

We prove asymptotic formulae for positive real $(p,q)$-Rogers-Szegő polynomials when the quadratic term in the expansion about the middle coefficient vanishes. For coefficient indices whose distance from $n/2$ is of order $n^{3/4}$, the coefficient ratios have a quadratic-quartic exponential limit. A uniform bound valid for all indices allows the ratios to be summed and gives locally uniform convergence of the centred generating polynomials to a Fourier--Laplace integral. We also obtain an asymptotic formula for the sum of the coefficients, convergence of all rescaled moments, weak convergence of the normalised coefficient measures, and convergence with multiplicity of the zeros tending to $w=1$.

math.CA↗

arXiv2023

Complex branches of a generalised Lambert $W$ function arising from $p,q$--binomial coefficients

The $ψ(x)$-function, which solves the equation $x = \sinh(aw)e^w$ for $0<a<1$, has a natural connection to the renowned Lambert $W$ function and also physical relevance through its connection to the Lenz-Ising model of ferromagnetism. We give a detailed analysis of its complex branches and construct Riemann surfaces from these under various conditions of $a$, unveiling intriguing new links to the Lambert $W$ function.

math.CV↗

arXiv2023

On a generalised Lambert $W$ branch transition function arising from $p,q$-binomial coefficients

With only a complete solution in dimension one and partially solved in dimension two, the Lenz-Ising model of magnetism is one of the most studied models in theoretical physics. An approach to solving this model in the high-dimensional case ($d>4$) is by modelling the magnetisation distribution with $p,q$-binomial coefficients. The connection between the parameters $p,q$ and the distribution peaks is obtained with a transition function $ω$ which generalises the mapping of Lambert $W$ function branches $W_0$ and $W_{-1}$ to each other. We give explicit formulas for the branches for special cases. Furthermore, we find derivatives, integrals, parametrizations, series expansions, and asymptotic behaviors.

math-ph↗