Identities for combinatorial sums involving trigonometric functions

Abstract

Let $$A_{m,n}(a)=\sum _{j=0}^m(-4{)}^j(\genfrac{}{}{0ex}{}{m+j}{2j})\sum _{k=0}^{n-1}\sin (a+2k\pi /n){\cos }^{2j}(a+2k\pi /n)$$ and $$B_{m,n}(a)=\sum _{j=0}^m(-4{)}^j(\genfrac{}{}{0ex}{}{m+j+1}{2j+1})\sum _{k=0}^{n-1}\sin (a+2k\pi /n){\cos }^{2j+1}(a+2k\pi /n),$$ where $m\ge 0$ and $n\ge 1$ are integers and $a$ is a real number. We present two proofs for the following results: (i) If $2m+1 \equiv 0 \, (\mbox{mod} \, n)$, then $$A_{m,n}(a)=(-1{)}^{m}n\sin ((2m+1)a).$$ (ii) If $2m+1 \not\equiv 0 \, (\mbox{mod} \, n)$, then $A_{m,n}(a)=0$. (iii) If $2(m+1) \equiv 0 \, (\mbox{mod} \, n)$, then $$B_{m,n}(a)=(-1{)}^m\frac{n}{2}\sin (2(m+1)a).$$ (iv) If $2(m+1) \not\equiv 0 \, (\mbox{mod} \, n)$, then $B_{m,n}(a)=0$.