I have a density function $f_n(z)$ with vanishing mean and unit variance. It obeys a recursive integral relation:

\(\displaystyle f_{n+1}(z) = \sqrt{(n+1)/n}\int_{-\infty}^{+\infty}f_n(z_n)f_0(\sqrt{n+1}z - \sqrt{n}z_n) dz_n\)

I wish to find a reasonable method to continue n onto the positive reals by interpolating $f_x(z)$ for any real $x$ between $n$ and $m$ such that $f_x(z)$ satisfies the recursion relation when $x$ is an integer. I have no idea whether there is a simple answer. Any suggestions would be gratefully appreciated.

P.S. Obviously, the interpolation must preserve normalization.

\(\displaystyle f_{n+1}(z) = \sqrt{(n+1)/n}\int_{-\infty}^{+\infty}f_n(z_n)f_0(\sqrt{n+1}z - \sqrt{n}z_n) dz_n\)

I wish to find a reasonable method to continue n onto the positive reals by interpolating $f_x(z)$ for any real $x$ between $n$ and $m$ such that $f_x(z)$ satisfies the recursion relation when $x$ is an integer. I have no idea whether there is a simple answer. Any suggestions would be gratefully appreciated.

P.S. Obviously, the interpolation must preserve normalization.

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