From a symmetry between $e$ and $π$ to the holonomic triangle
In 2002, the author studied together the recurrences $u_{n+2}=u_{n+1}+u_n/n$ and $v_{n+2}=v_{n+1}/n+v_n$, in which the factor $1/n$ is placed in a symmetric way. The first one leads to $e$ and the second one to $π$, and each of them is classical when taken separately. When $n$ is replaced by $n+m$ in the denominators, with an integer $m\ge0$, the solutions grow like $f(m)\,n$ and $g(m)\sqrt n$, and the connection constants $f(m)$ and $g(m)$, suitably normalized, are explicit linear forms in $1$ and $e$ and in $1$ and $π$. We let $m$ become a complex variable $z$. Then $f$ and $g$ become meromorphic functions of $z$, the connection functions, which satisfy linear functional equations of order two in $z$, as the Gamma function satisfies $Γ(z+1)=zΓ(z)$. They add a third object to the classical pair formed by a P-recursive sequence and its D-finite generating function. We characterize them by their functional equations and their growth, as Wielandt's theorem does for the Gamma function, we show that they are differentially transcendental, and we explain the linear forms at the integers by the solutions excluded by the growth condition. In general, when the coefficients of a linear recurrence in $n$ depend on the parameter only through $n+z$, a translation argument gives a functional equation for the connection constant under the normalization and convergence assumptions of our translation theorem. Conversely, a solution of a linear functional equation with polynomial coefficients which has uniform polynomial growth in a right half-plane is the limit of an explicit P-recursive family of interpolation polynomials, and this applies to all the connection functions given by our existence theorem for coefficients of degree one in $n+z$.