In the paper, with the help of Kazarinoff's integral representation for the ratio of two gamma functions, with the aid the duplication formula of the digamma function, by virtue of integral ...representations of polygamma functions, and in the light of the L'Hôpital type monotonicity rule, the author describes extended binomial coefficients in terms of the gamma functions and the falling factorials, collects three integral representations of central binomial coefficients, establishes three integral representations of extended central binomial coefficients, proves decreasing and increasing properties of two power‐exponential functions involving extended (central) binomial coefficients, and derives several double and sharp inequalities for bounding extended (central) binomial coefficients, and compares these inequalities with known ones.
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 W0 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 behaviours.
•Builds on the connection between the Ising model and p,q-binomial distribution.•Defines a generalization of the Lambert W branch transition function.•Explicit formulas and values of the new functions are given in some special cases.•We give derivatives and primitive functions, including some definite integrals.•We obtain bounds, series expansions, and asymptotes using novel techniques.
We discuss the integer sequence transform a 1→ b, where bn is the number of real roots of the polynomial a0 + a1x + a2x2 + · · · + anxn. It is shown that several sequences a give the trivial sequence ...b = (0, 1, 0, 1, 0, 1, . . .), i.e., bn = n mod 2, among them the Catalan numbers, central binomial coefficients, n! and for a fixed k. We also look at some sequences a for which b is more interesting such as an = (n + 1)k for k ≥ 3. Further, general procedures are given for constructing real sequences an for which bn is either always maximal or minimal.
We discuss
q
-analogues of the classical congruence
ap
b
p
≡
a
b
(
mod
p
3
)
, for primes
p
>
3
, as well as its generalisations. In particular, we prove related congruences for (
q
-analogues of) ...integral factorial ratios.
In this note, we shall give conditions which guarantee that 1-qb1-qanm∈Zq holds. We shall provide a full characterisation for 1-qb1-qakam∈Zq. This unifies a variety of results already known in ...literature. We shall prove new divisibility properties for the binomial coefficients and a new divisibility result for a certain finite sum involving the roots of the unity.
We establish a q-analogue of Sun–Zhao's congruence on harmonic sums. Based on this q-congruence and a q-series identity, we prove a congruence conjecture on sums of central q-binomial coefficients, ...which was recently proposed by Guo. We also deduce a q-analogue of a congruence due to Apagodu and Zeilberger from Guo's q-congruence.
By examining two hypergeometric series transformations, we establish several remarkable infinite series identities involving harmonic numbers and quintic central binomial coefficients, including five ...conjectured recently by Z.-W. Sun ‘Series with summands involving harmonic numbers’, Preprint, 2023, arXiv:2210.07238v7. This is realised by ‘the coefficient extraction method’ implemented by Mathematica commands.
Abstract
We prove some congruences on sums involving fourth powers of central
q
-binomial coefficients. As a conclusion, we confirm the following supercongruence observed by Long Pacific J. Math. 249 ...(2011), 405–418:
$$\sum\limits_{k = 0}^{((p^r-1)/(2))} {\displaystyle{{4k + 1} \over {{256}^k}}} \left( \matrix{2k \cr k} \right)^4\equiv p^r\quad \left( {\bmod p^{r + 3}} \right),$$
where
p
⩾5 is a prime and
r
is a positive integer. Our method is similar to but a little different from the WZ method used by Zudilin to prove Ramanujan-type supercongruences.
We prove a general combinatorial formula involving the reciprocals of the binomial coefficients and the partial sum of an arbitrary sequence. Applying this formula we offer many combinatorial ...identities involving reciprocals of the binomial and central binomial coefficients and harmonic numbers; some of which have been considered previously and the others are new.
In a 2023 Discrete Mathematics article, Mneimneh introduced a remarkable formula for a binomial sum of harmonic numbers, defined by Hk=∑i=1k1/i. This formula can be extended to the generalized ...harmonic numbers, where 1/i is replaced by 1/ir for a positive integer r. In this paper, we extend Mneimneh's formula to the generalized hyperharmonic numbers hk(ℓ,r), and as a special case of this, to the generalized harmonic numbers when ℓ=1. Some recent notes by Campbell in a follow-up 2023 Discrete Mathematics article are also considered.
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