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{\bf{\centerline{Homework \#2 Mathematics 8669 Due Monday February 29, 2016}}}
\vskip10pt\noindent
1. Verify the following identities using hypergeometric series.
\begin{equation}
\sum_{k=0}^n (-1)^k\binom{n}{k}\frac{x}{x+k}=1/\binom{x+n}{n}.
\end{equation}

\begin{equation}
\sum_{k=1}^n (-1)^{k-1}\binom{n}{k}\frac{1}{k}=
\sum_{k=1}^n \frac{1}{k}.
\end{equation}

\begin{equation}
\sum_{k=0}^{n/2} (-1)^k\binom{n-k}{k}2^{n-2k}
=n+1.
\end{equation}

\begin{equation}
\sum_{k=r}^{n-s} \binom{k}{r}\binom{n-k}{s}=\binom{n+1}{r+s+1}.
\end{equation}


\begin{equation}
\sum_{k=0}^r \binom{x}{k}\binom{-x}{n-k}=\frac{n-r}{n}\binom{x-1}{r}
\binom{-x}{n-r} \quad (\text{here }r\le n).
\end{equation}


\begin{equation}
\sum_{k=0}^{n/2} \binom{x}{k}\binom{x-k}{n-2k} \frac{1}{2^{2k}}=2^{-n}\binom{2x}{n}.
\end{equation}

\begin{equation} 
\sum_{k=0}^{n} \binom{n}{k}\binom{n+2r+k}{n+r}(-1)^k2^{n-k}=
(-1)^{n/2}\frac{1+(-1)^n}{2}\frac{\binom{n+r}{n/2}\binom{n+2r}{r}}
{\binom{n+r}{r}}.
\end{equation}

\begin{equation}
\sum_{k=0}^{n} \binom{n}{k}\binom{k/2}{m}=
\frac{n}{m}\binom{n-m-1}{m-1}2^{n-2m}.
\end{equation}

\begin{equation}
\sum_{k=0}^n \binom{m-x+y}{k}\binom{n+x-y}{n-k}\binom{x+k}{m+n}=
\binom{x}{m}\binom{y}{n}.
\end{equation}

\begin{equation}
\sum_{k=1}^n (-1)^k\binom{n}{k}\binom{n+k-1}{k}\sum_{j=1}^k \frac{1}{j}=
\frac{(-1)^n}{n}.
\end{equation}

\begin{equation}
\sum_{k=0}^{2n} \frac{\binom{2n}{k}(-1)^k}{\binom{2n+2x}{k+x}\binom{2n+2y}{k+y}}=
\frac{\binom{2n}{n}\binom{2n+x+y+1}{n}}{\binom{x+n}{n}\binom{y+n}{n}\binom{2n+2x}{n+x}
\binom{2n+2y}{n+y}}.
\end{equation}

\begin{equation}
\sum_{k=0}^{2n} (-1)^{n-k}\binom{2n}{k}^3=
\frac{(3n)!}{n!n!n!}.
\end{equation}

\vskip10pt\noindent
2. Expand $(1-x)^A$ in terms of powers of $x/(1-x)^2$ by Lagrange inversion.
Then evaluate
$$
\ _2F_1\left( \left. \begin{matrix}
a,&a+1/2\\
&2a
\end{matrix}  \right| \frac{-4x}{(1-x)^2}\right), 
\quad\ _2F_1\left( \left. \begin{matrix}
a,&a+1/2\\
&2a+1
\end{matrix}  \right| \frac{-4x}{(1-x)^2}\right).
$$
How is this related to the Catalan number generating function
$$
\sum_{n=0}^\infty \frac{1}{n+1}\binom{2n}{n} t^n?
$$
\vskip 10 pt\noindent 
3. Let $a_1, a_2, a_3$ be non-negative integers. 
Prove that the constant term of the Laurent polynomial
$$
 \prod_{1\le i\neq j\le 3} (1-x_i/x_j)^{a_i} {\text{\quad is \quad}} 
 \binom{a_1+a_2+a_3}{a_1,a_2,a_3}.
$$
\vskip 10 pt\noindent 
4. Let $A$ and $B$ be relatively prime positive integers. What is
the coefficient of $z^{AB}$ in the power series for  
$$
\frac{(1-z^{A+B})^{A+B}}{(1-z^A)^A (1-z^B)^B}?
$$
Do you need to use $GCD(A,B)=1$?
\vskip 10 pt\noindent 
5. Find a product formula for the sum
$$
\sum_{k=-n}^n \left[\begin{matrix} 2n\\n-k\end{matrix}\right]_q
q^{\binom{k}{2}} x^k.
$$
What happens if $n\to \infty$?
\vskip 10 pt\noindent 
6. Using weighted integer partitions, give a bijective proof of
$$
(b+aq)\sum_{n=0}^\infty \frac{(-aq;q)_n}{(bq;q)_n} q^n=\frac{(-aq;q)_\infty}{(bq;q)_\infty}-(1-b).
$$


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