arXiv subjects

Karl Dilcher

Publications and source records attributed to Karl Dilcher.

At least 19 recordsLinked to original sources

arXiv2026

Connections between colored restricted $b$-ary and ordinary partitions

We establish various connections between classes of colored and bounded ordinary partitions on one hand, and colored but not necessarily bounded binary and $b$-ary partitions on the other hand. Many of these results are based on special recurrence relations, some of which are new. We also obtain several classes of identities for sequences of colored $b$-ary partitions, and there are a few results concerning compositions.

math.CO↗

arXiv2026

New central $q$-binomial identities

We establish several new series evaluations involving the central $q$-binomial coefficients, with the inspiration coming from earlier work by Vignat and one of the authors on the limiting case at $q\to 1$.

math.CO↗

arXiv2026

Properties of two Chebyshev-like polynomial sequences

By modifying the generating function of the Chebyshev polynomials of the second kind, we obtain a sequence of reciprocal (or palindromic) polynomials, as well as a related companion sequence. Among numerous other properties, we obtain discriminant and resultant identities for these polynomial sequences and prove partial irreducibility results. Throughout, we point to parallels and connections with the Chebyshev polynomials of both kinds.

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arXiv2025

Integrals involving arbitrary powers of the arcsine, with applications to infinite series

Using appropriate power series evaluations, we determine all moments of arbitrary positive powers of the arcsine. As consequences we evaluate several doubly infinite classes of power series involving central binomial coefficients and generalized multiple harmonic sums. By specializing the variable involved, we then evaluate classes of numerical sequences, mostly in terms of powers of $\pi$. Finally, we obtain limit expressions for arbitrary powers of $\pi$.

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arXiv2025

Powers of the arcsine and infinite classes of series involving central binomial coefficients

A general integral expression to transform power series is applied to $\arcsin{x}$ and its positive integer powers. We concentrate on the first to the fourth powers and obtain infinite classes of new power series involving central binomial coefficients. Specializing the variable to appropriate simple values leads to different classes of series expansions for $\pi$ and some of its positive integer powers. We also discuss several limit expressions and connections with hypergeometric series.

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arXiv2025

An alternating sum of the floor function of square roots

We show that the alternating sum of the floor function of $\sqrt{jn}$, with $j$ ranging from 1 to $n$, has an easy evaluation for all odd integers $n\geq 1$. This is in contrast to known non-alternating sums of the same type which hold only for a class of primes. The proof is elementary and was suggested by an AI model. To put this result in perspective, we also prove an asymptotic expression for the analogous sum without the floor function.

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arXiv2025

Further Classes of Series Involving Central Binomial Coefficients

Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including exponential Bell polynomials and integral representations, to further extend these results. Throughout the paper, we make extensive use of the gamma and polygamma functions and their properties.

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arXiv2025

Sums of the floor function related to class numbers of imaginary quadratic fields

A curious identity of Bunyakovsky (1882), made more widely known by P\'olya and Szeg{\H o} in their ``Problems and Theorems in Analysis", gives an evaluation of a sum of the floor function of square roots involving primes $p\equiv 1\pmod{4}$. We evaluate this sum also in the case $p\equiv 3\pmod{4}$, obtaining an identity in terms of the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-p})$. We also consider certain cases where the prime $p$ is replaced by a composite integer. Class numbers of imaginary quadratic fields are again involved in some cases.

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arXiv2025

Colored base-3 partitions, sequences of polynomials, and perfect numbers

Motivated by the observation that the counting function of a certain base-3 colored partition contains the even perfect numbers as a subsequence, we begin by defining a sequence of polynomials in four variables and discuss their properties and combinatorial interpretations. We then concentrate on certain subsequences that are related to the Chebyshev polynomials of both kinds. Finally, we consider several sequences of single-variable polynomials that have meaningful combinatorial interpretations as well as interesting zero distributions.

math.CO↗

arXiv2024

Polynomials and algebraic curves related to certain binary and $b$-ary overpartitions

We begin by considering a sequence of polynomials in three variables whose coefficients count restricted binary overpartitions with certain properties. We then concentrate on two specific subsequences that are closely related to the Chebyshev polynomials of both kinds, deriving combinatorial and algebraic properties of some special cases. We show that the zeros of these polynomial sequences lie on certain algebraic curves, some of which we study in greater detail. Finally, we extend part of this work to restricted $b$-ary overpartitions for arbitrary integers $b\geq 2$.

math.CO↗

arXiv2021

Divisibility and Arithmetic Properties of a Class of Sparse Polynomials

We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider rational roots. This leads to the study of an infinite class of integer sequences which have interesting properties and satisfy linear recurrence relations.

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arXiv2021

A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence

The Stern polynomials defined by $s(0;x)=0$, $s(1;x)=1$, and for $n\geq 1$ by $s(2n;x)=s(n;x^2)$ and $s(2n+1;x)=x\,s(n;x^2)+s(n+1;x^2)$ have only 0 and 1 as coefficients. We construct an infinite lower-triangular matrix related to the coefficients of the $s(n;x)$ and show that its inverse has only 0, 1, and $-1$ as entries, which we find explicitly. In particular, the sign distribution of the entries is determined by the Prouhet-Thue-Morse sequence. We also obtain other properties of this matrix and a related Pascal-type matrix that involve the Catalan, Stirling, Fibonacci, Fine, and Padovan numbers. Further results involve compositions of integers, the Sierpiński matrix, and identities connecting the Stern and Prouhet-Thue-Morse sequences.

math.CO↗

arXiv2021

Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Hankel determinants of sequences related to Bernoulli and Euler numbers have been studied before, and numerous identities are known. However, when a sequence is shifted by one unit, the situation often changes significantly. In this paper we use classical orthogonal polynomials and related methods to prove a general result concerning Hankel determinants for shifted sequences. We then apply this result to obtain new Hankel determinant evaluations for a total of $13$ sequences related to Bernoulli and Euler numbers, one of which concerns Euler polynomials.

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arXiv2020

On a result of Koecher concerning Markov-Apéry type formulas for the Riemann zeta function

Koecher in 1980 derived a method for obtaining identities for the Riemann zeta function at odd positive integers, including a classical result for $ζ(3)$ due to Markov and rediscovered by Apéry. In this paper we extend Koecher's method to a very general setting and prove two more specific but still rather general results. As applications we obtain infinite classes of identities for alternating Euler sums, further Markov-Apéry type identities, and identities for even powers of $π$

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arXiv2020

Some Properties of a Class of Sparse Polynomials

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity and log-concavity, as well as identities involving derivatives. We also prove upper and lower bounds on the moduli of the zeros of these polynomials.

math.CA↗

arXiv2020

Hankel Determinants of sequences related to Bernoulli and Euler Polynomials

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials.

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arXiv2020

Orthogonal polynomials and Hankel Determinants for certain Bernoulli and Euler Polynomials

Using continued fraction expansions of certain polygamma functions as a main tool, we find orthogonal polynomials with respect to the odd-index Bernoulli polynomials $B_{2k+1}(x)$ and the Euler polynomials $E_{2k+ν}(x)$, for $ν=0, 1, 2$. In the process we also determine the corresponding Jacobi continued fractions (or J-fractions) and Hankel determinants. In all these cases the Hankel determinants are polynomials in $x$ which factor completely over the rationals.

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arXiv2019

Arithmetic properties of polynomial solutions of the Diophantine equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$

For each integer $n\geq 1$ we consider the unique polynomials $P, Q\in\mathbb{Q}[x]$ of smallest degree $n$ that are solutions of the equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$. We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility results. We also consider some related polynomials and their properties.

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