(1)

\begin{align*} \frac{1}{1-ax-bx^{2}} & =\frac{1}{1-\left(ax+bx^{2}\right)}\\ & =\sum_{k=0}^{\infty}\left(ax+bx^{2}\right)^{k}\\ & =\sum_{k=0}^{\infty}\sum_{j=0}^{k}C(k,j)\left(ax\right)^{j}\left(bx^{2}\right)^{k-j}\\ & =\sum_{k=0}^{\infty}\sum_{j=0}^{k}C(k,j)a^{j}b^{k-j}x^{2k-j}\\ & =\sum_{s=0}^{\infty}\sum_{t=s}^{2s}C(s,2s-t)a^{2s-t}b^{t-s}x^{t}\cmt{s=k\;,\;t=2k-j}\\ & =\sum_{s=0}^{\infty}\sum_{t=s}^{2s}C(s,t-s)a^{2s-t}b^{t-s}x^{t}\\ & =\sum_{t=0}^{\infty}\sum_{s=\left\lceil \frac{t}{2}\right\rceil }^{t}C(s,t-s)a^{2s-t}b^{t-s}x^{t}\\ & =\sum_{t=0}^{\infty}\sum_{s=0}^{\left\lfloor \frac{t}{2}\right\rfloor }C(t-s,s)a^{t-2s}b^{s}x^{t} \end{align*}

(2)

\begin{align*} \frac{1}{1-\left(\alpha+\beta\right)x+\alpha\beta x^{2}} & =\frac{1}{\left(1-\alpha x\right)\left(1-\beta x\right)}\\ & =\frac{1}{\beta-\alpha}\left(\frac{\beta}{1-\beta x}-\frac{\alpha}{1-\alpha x}\right)\\ & =\frac{1}{\beta-\alpha}\left(\sum_{k=0}^{\infty}\beta\left(\beta x\right)^{k}-\sum_{k=0}^{\infty}\alpha\left(\alpha x\right)^{k}\right)\\ & =\frac{1}{\beta-\alpha}\sum_{k=0}^{\infty}\left(\beta^{k+1}-\alpha^{k+1}\right)x^{k}\\ & =\sum_{k=0}^{\infty}\frac{\beta^{k+1}-\alpha^{k+1}}{\beta-\alpha}x^{k} \end{align*}

(3)

\begin{align*} \sum_{k-0}^{\infty}\left(a^{k}+b^{k}\right)x^{k} & =\sum_{k-0}^{\infty}\left(a^{k}x^{k}+b^{k}x^{k}\right)\\ & =\frac{1}{1-ax}+\frac{1}{1-bx}\\ & =\frac{2-\left(a+b\right)x}{1-\left(a+b\right)x+abx^{2}}\\ & =\left\{ 2-\left(a+b\right)x\right\} \sum_{k=0}^{\infty}\sum_{j=0}^{\left\lfloor \frac{k}{2}\right\rfloor }C\left(k-j,j\right)\left(a+b\right)^{k-2j}\left(-ab\right)^{j}x^{k}\\ & =2\sum_{k=0}^{\infty}\sum_{j=0}^{\left\lfloor \frac{k}{2}\right\rfloor }C\left(k-j,j\right)\left(a+b\right)^{k-2j}\left(-ab\right)^{j}x^{k}-\sum_{k=0}^{\infty}\sum_{j=0}^{\left\lfloor \frac{k}{2}\right\rfloor }C\left(k-j,j\right)\left(a+b\right)^{k-2j+1}\left(-ab\right)^{j}x^{k+1}\\ & =2\sum_{k=0}^{\infty}\sum_{j=0}^{\left\lfloor \frac{k}{2}\right\rfloor }C\left(k-j,j\right)\left(a+b\right)^{k-2j}\left(-ab\right)^{j}x^{k}-\sum_{k=1}^{\infty}\sum_{j=0}^{\left\lfloor \frac{k}{2}\right\rfloor }C\left(k-j,j\right)\left(a+b\right)^{k-2j}\left(-ab\right)^{j}x^{k}\\ & =2\sum_{k=1}^{\infty}\sum_{j=0}^{\left\lfloor \frac{k}{2}\right\rfloor }C\left(k-j,j\right)\left(a+b\right)^{k-2j}\left(-ab\right)^{j}x^{k}+2-\sum_{k=1}^{\infty}\sum_{j=0}^{\left\lfloor \frac{k}{2}\right\rfloor }\frac{k-2j}{k-j}C\left(k-j,j\right)\left(a+b\right)^{k-2j}\left(-ab\right)^{j}x^{k}\\ \\ & =\sum_{k=1}^{\infty}\sum_{j=0}^{\left\lfloor \frac{k}{2}\right\rfloor }\frac{2k-2j-\left(k-2j\right)}{k-j}C(k-j,j)\left(a+b\right)^{k-2j}\left(-ab\right)^{j}x^{k}+2\\ & =\sum_{k=1}^{\infty}\sum_{j=0}^{\left\lfloor \frac{k}{2}\right\rfloor }\left(-1\right)^{j}\frac{k}{k-j}C(k-j,j)\left(a+b\right)^{k-2j}\left(ab\right)^{j}x^{k}+2 \end{align*} これより、\(x\)の係数を比較して
\[ a^{k}+b^{k}=\left(1-\delta_{0,k}\right)\sum_{j=0}^{\left\lfloor \frac{k}{2}\right\rfloor }\left(-1\right)^{j}\frac{k}{k-j}C\left(k-j,j\right)\left(a+b\right)^{k-2j}\left(ab\right)^{j}+2\delta_{0,k} \]

(4)

\begin{align*} T_{n}\left(\cos\theta\right) & =\cos\left(n\theta\right)\\ & =\frac{e^{in\theta}+e^{-in\theta}}{2}\\ & =\frac{1}{2}\left\{ \left(e^{i\theta}\right)^{n}+\left(e^{-i\theta}\right)^{n}\right\} \\ & =\frac{1}{2}\left(\left(1-\delta_{0,n}\right)\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(-1\right)^{j}\frac{n}{n-j}C\left(n-j,j\right)\left(e^{i\theta}+e^{-i\theta}\right)^{n-2j}\left(e^{i\theta}e^{-i\theta}\right)^{j}+2\delta_{0,n}\right)\\ & =\left(1-\delta_{0,n}\right)\sum_{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(-1\right)^{j}\frac{n}{n-j}C\left(n-j,j\right)2^{n-2j-1}\cos^{n-2j}\theta+\delta_{0,n} \end{align*}