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I would like to find the closed form (if there exists one) of the two recurrence relations \begin{align*} a_n&=(\gamma-n+1)a_{n-1}+\omega b_{n-1}\\ b_n&=(\gamma-n+1)b_{n-1}-\omega a_{n-1}, \end{align*} with the initial conditions $a_0=1$ and $b_0=0$, where $\gamma\in(0,1)$ and $\omega>0$. The first few terms of the sequences are \begin{align*} a_0&=1 & b_0&=0\\ a_1&=\gamma & b_1&=-\omega\\ a_2&=\gamma(\gamma-1)-\omega^2 & b_2&=-\omega(\gamma-1)-\omega\gamma\\ \end{align*}

I've tried using summations to find the solution. Let $A(x)=\sum_{n=0}^\infty a_nx^n$ and $B(x)=\sum_{n=0}^\infty b_nx^n$. Then

\begin{align*} \sum_{n=1}^\infty a_nx^n&=\sum_{n=1}^\infty(\gamma-n+1)a_{n-1}x^n+\sum_{n=1}^\infty\omega b_{n-1}x^n\\ &=\gamma\sum_{n=1}^\infty a_{n-1}x^n-\sum_{n=1}^\infty(n-1)a_{n-1}x^n+\omega\sum_{n=1}^\infty b_{n-1}x^n\\ &=\gamma x\sum_{n=0}^\infty a_nx^n-x^2\sum_{n=0}^\infty na_nx^{n-1}+\omega\sum_{n=0}^\infty b_nx^n\\ &=\gamma xA(x)-x^2A'(x)+\omega B(x). \end{align*} which implies $$ A(x)-1=x\gamma A(x)-x^2A'(x)+\omega xB(x). $$

Similarly, we have $$B(x)=x\gamma B(x)-x^2B'(x)-\omega xA(x).$$

We can then use substitution to get an extremely nasty second-order differential equation. Is there any way to go from here, or is there another method that is known on how to solve recurrence relations like these?

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    $\begingroup$ Take the first equation, solve it for $b_{n-1}$, substitute into the second equation to get a recurrence involving only the $a$-sequence. $\endgroup$ – Gerry Myerson Nov 20 '17 at 1:46
  • $\begingroup$ @GerryMyerson Oh my. Of course, thank you! Feel free to leave that as an answer. $\endgroup$ – Bonnaduck Nov 20 '17 at 2:14
  • $\begingroup$ Let me encourage you to write it up, with details, and then to post it as an answer. $\endgroup$ – Gerry Myerson Nov 20 '17 at 2:47
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Notice that $$\omega b_{n-1}=a_n-(\gamma-n+1)a_{n-1}.$$ Substituting into the second equation yields the recurrence relation \begin{align}a_n=(2\gamma-2n+3)a_{n-1}-((\gamma-n+2)^2+\omega^2)a_{n-2}, \tag{$\ast$}\end{align} with $a_0=1$ and $a_1=\gamma$.

Using generating function $A(x)=\sum_{n=0}^\infty a_nx^n$, we get the differential equation $$\small{ x^4A''(x)+(x^3-2\gamma x^2+2x)A'(x)+((\gamma^2+\omega^2)x^2-(2\gamma+1)x+1)A(x)+2(\gamma+1)x-(1+\gamma)=0}.$$

This is beyond my abilities to solve. However, Wolfram Alpha solves ($\ast$), giving

$$a_n=\frac12(-1)^n\left(\frac{\Gamma(n-(\gamma+i\omega))}{\Gamma(-(\gamma+i\omega))}+\frac{\Gamma(n-(\gamma-i\omega))}{\Gamma(-(\gamma-i\omega))}\right).$$ Since $n\in\mathbb{N}$, we get \begin{align*} a_n&=\frac12((\gamma+i\omega)_n+(\gamma-i\omega)_n)\\ &=\frac12\sum_{k=0}^n\left[s(n,k)(\gamma+i\omega)^k+s(n,k)(\gamma-i\omega)^k\right]\\ &=\sum_{k=0}^ns(n,k)r^k\cos(k\theta), \end{align*} where $(\cdot)_n$ is the falling factorial, $s(\cdot,\cdot)$ is the Stirling number of the first kind, and we use the substitution $\gamma=r\cos\theta$ and $\omega=r\sin\theta$. Alternatively, without the trig substitution, we can get $$a_n=\sum_{k=0}^n s(n,k)\sum_{m=0}^k {k\choose m}\gamma^{k-m}\omega^m\cos\left(\frac{m\pi}{2}\right)$$

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