Quadrinomial-Like Versions for Wolstenholme, Morley and Glaisher Congruences

The quadrinomial coefficient is defined as the coefficient of $x^k$ in the polynomial expansion of $\left( 1+x+x^2+x^3\right) ^{n}$, where $n$ and $k$ are nonnegative integers. In the present paper, we derive some congruences involving the quadrinomial coefficients. For instance, we establish two congruences that are analogous to those of Morley and Wolstenholme.

Keywords

Fermat quotient, Binomial coefficients; factorials; $q$-identities, quadrinomial coefficient, Taylor expansion, Congruences; primitive roots; residue systems, Bernoulli and Euler numbers and polynomials, Euler numbers, Factorials, binomial coefficients, combinatorial functions, Wolstenholme's congruence, Pell sequence

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