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 $m >2$.
Mikic,J . (2026). On New Divisibility Properties of Generalized Central Trinomial Coefficients and Legendre Polynomials. Transactions in Theoretical and Mathematical Physics, (), 90-103. doi: 10.30511/ttmp.2026.2093603.1082
MLA
Mikic,J . "On New Divisibility Properties of Generalized Central Trinomial Coefficients and Legendre Polynomials", Transactions in Theoretical and Mathematical Physics, , , 2026, 90-103. doi: 10.30511/ttmp.2026.2093603.1082
HARVARD
Mikic J. (2026). 'On New Divisibility Properties of Generalized Central Trinomial Coefficients and Legendre Polynomials', Transactions in Theoretical and Mathematical Physics, (), pp. 90-103. doi: 10.30511/ttmp.2026.2093603.1082
CHICAGO
J Mikic, "On New Divisibility Properties of Generalized Central Trinomial Coefficients and Legendre Polynomials," Transactions in Theoretical and Mathematical Physics, (2026): 90-103, doi: 10.30511/ttmp.2026.2093603.1082
VANCOUVER
Mikic J. On New Divisibility Properties of Generalized Central Trinomial Coefficients and Legendre Polynomials. TTMP. 2026;():90-103. doi: 10.30511/ttmp.2026.2093603.1082
