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In this paper, by using the tool of trinomial coefficients we study some determinant problems posed by Zhi-Wei Sun. For example, given any odd prime $p$ with $pequiv 2pmod 3$, we show that $2det[frac{1}{i^2-ij+j^2}]_{1le i,jle p-1}$ is a quadratic residue modulo $p$. This confirms a conjecture of Zhi-Wei Sun.
The discriminant of a polynomial of the form $pm x^n pm x^m pm 1$ has the form $n^n pm m^m(n-m)^{n-m}$ when $n,m$ are relatively prime. We investigate when these discriminants have prime power divisors. We explain several symmetries that appear in th
We establish a congruence on sums of central $q$-binomial coefficients. From this $q$-congruence, we derive the divisibility of the $q$-trinomial coefficients introduced by Andrews and Baxter.
We show that the formalism of hybrid polynomials, interpolating between Hermite and Laguerre polynomials, is very useful in the study of Motzkin numbers and central trinomial coefficients. These sequences are identified as special values of hybrid po
We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an $L$-function. As immediate applicat
For a given sequence $mathbf{alpha} = [alpha_1,alpha_2,dots,alpha_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(mathbf{alpha})(t)$ that counts the nonnegative integer solutions of the equation $alpha_1x_1+alpha_2 x_2+c