ON BINOMIAL SUMS WITH THE TERMS OF SEQUENCES {g_kn} AND {h_kn}

Abstract

In this paper, we derive sums and alternating sums of products of terms of
the sequences $\left\{ g_{kn}\right\} $ and $\left\{ h_{kn}\right\} $ with
binomial coefficients. For example,
\begin{eqnarray*}
&\sum\limits_{i=0}^{n}\binom{n}{i}\left( -1\right) ^{i} \left(c^{2k}\left(-q\right) ^{k}+c^{k}v_{k}+1\right)^{-ai}h_{k\left( ai+b\right) }h_{k\left(ai+e\right) } \\
&=\left\{
\begin{array}{clc}
-\Delta ^{\left( n+1\right) /2}g_{k\left( an+b+e\right) }g_{ka}^{n}\left(
c^{2k}\left( -q\right) ^{k}+c^{k}v_{k}+1\right) ^{-an} & \text{if }n\text{ is odd,} & \\
\Delta ^{n/2}h_{k\left( an+b+e\right) }g_{ka}^{n}\left( c^{2k}\left(
-q\right) ^{k}+c^{k}v_{k}+1\right) ^{-an} & \text{if }n\text{ is even,} &
\end{array}%
\right.
\end{eqnarray*}%
and
\begin{eqnarray*}
&&\sum\limits_{i=0}^{n}\binom{n}{i}i^{\underline{m}}g_{k\left( n-ti\right)
}h_{kti} \\
&&=2^{n-m}n^{\underline{m}}g_{kn}-n^{\underline{m}}\left( c^{2k}\left(
-q\right) ^{k}+c^{k}v_{k}+1\right) ^{n\left( 1-t\right)
}h_{kt}^{n-m}g_{k\left( tm+tn-n\right) },
\end{eqnarray*}%
where $a, b, e$ is any integer numbers, $c$ is nonzero real number and $m$
is nonnegative integer.