Calculate determinant of a continuant matrix with variable elements?

Thanks to a comment by Jean-Claude Arbaut I finally got the answer from the page 559 of Muir's book (after correcting for some typos) $$ f_n(x,y)=\sum _{k=0}^n (-1)^{n-k} \binom{n}{k} \left(\frac{x+y-1}{2}\right)^{(k)} \left(\frac{-x+y-1}{2}\right)^{(n-k)}, $$ where $(a)^{(n)}=a(a-1)\ldots (a-n+1)$ denotes the falling factorial.

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Linear Algebra

Matrices

Tridiagonal Matrices

Determinant

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