Revista de la
Unión Matemática Argentina
Combinatorial and modular solutions of some sequences with links to a certain conformal map
Pablo A. Panzone
Volume 59, no. 2 (2018), pp. 389–414


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If $f_n$ is a free parameter, we give a combinatorial closed form solution of the recursion \[ (n+1)^2 u_ {n+1} -f_n u_n-n^2 u_ {n-1} =0, \quad n \geq 1, \] and a related generating function. This is used to give a solution to the Apéry type sequence \[ r_n n^3+r_ {n-1} \left \{ \alpha n^3- \frac{3\alpha}{2} n^2+ \left \{ \frac{\alpha}{2} +2 \theta \right \} n- \theta \right \} +r_ {n-2} (n-1)^3=0, \quad n \geq 2, \] for certain parameters $ \alpha, \theta$. We show from another viewpoint two independent solutions of the last recursion related to certain modular forms associated with a problem of conformal mapping: Let $f( \tau)$ be a conformal map of a zero-angle hyperbolic quadrangle to an open half plane with values $0$, $ \rho$, $1$, $ \infty$ ($0< \rho<1$) at the cusps and define $t=t( \tau): = \frac{1}{\rho} f( \tau) \frac{f(\tau)-\rho}{f(\tau)-1} $. Then the function \[ E( \tau)= \frac{1}{2\pi i} \frac{f'(\tau)}{f(\tau)} \frac{1}{1-\frac{f(\tau)}{\rho}} \] is a solution, as a generating function in the variable $t$, of the above recurrence. In other words, $E( \tau)=r_0+r_1t+r_2 t^2+ \dots$, where $r_0=1$, $r_1=- \theta$, $ \alpha=2- \frac{4}{\rho} $.