Abstract
Let us choose a positive integern and denoteF(x, y)=\(\sum _{m = 0}^n f^{(n - m)} (x)g^{(m)} (y)\), wheref(·) andg(·) are arbitrary sufficiently smooth functions. Three different proofs of the validity of the relation are given. We also establish discrete and noncommutative analogs of this identity.
$$F(x, y) - F(y, x) = \int_y^x {\{ f^{(n + 1)} (t)g(x + y - t) - f(t)g^{(n + 1)} (x + y - t)\} dt.} $$
Key words
Darboux’s identity Appell polynomials Poisson-Charlier polynomials Rodrigues’ formulaPreview
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© Kluwer Academic/Plenum Publishers 2000