Transactions in Theoretical and Mathematical Physics

Transactions in Theoretical and Mathematical Physics

On New Divisibility Properties of Generalized Central Trinomial Coefficients and Legendre Polynomials

Document Type : Original Article

Author
Faculty of Technology University of Banja Luka Bosnia and Herzegovina
Abstract
We present a new formula for the highest power of $a+b$ that divides the sum $B(n,m,a,b)=\sum_{k=0}^{n}\binom{n}{k}^m a^{n-k}b^k$ for the case $m=2$. By using this formula, we give a complete 3-adic valuation for central Dellanoy numbers.
Also, we find the highest power of an odd integer $x$ that divides Legendre's polynomial $P_{n}(x)$. By using the same idea, generalized trinomial coefficients and generalized Motzkin numbers are treated. As a result, we give a complete 3-adic valuation for little Schr\"{o}der numbers and restricted hexagonal numbers.
By using a new class of binomial sums, we examine the divisibility of $B(n,m,a,b)$ by powers of $a+b$ for $m >2$.
Keywords

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Volume 3, Issue 3
Summer 2026
Pages 90-103

  • Receive Date 06 July 2026
  • Accept Date 31 July 2026
  • First Publish Date 01 August 2026
  • Publish Date 01 August 2026