Newton Identities for Weierstrass Products

We prove a generalization of the Newton Identities for entire functions, which give a relation between the Taylor coefficients and sums of powers of reciprocals of the zeros of an entire function. We apply these identities to a number of special functions, yielding some interesting recursion relations. By the Weierstrass Product Theorem, any entire function f : C → C can be written as a normally convergent product f(z) = ez ∏ n≥1 [( 1− z rn ) exp ( z rn + 1 2 ( z rn )2 + · · ·+ 1 kn ( z rn )kn)] , (1) where g is an entire function, (rn)n≥1 is the finite or infinite (or empty) sequence of non-zero complex roots of f , m ∈ Z≥0 is the order of vanishing of f at 0, and kn ∈ Z≥0 for each n. This, and the special functions considered below, can be found, for example, in [1]. The normal convergence of (1) in C is equivalent to the absolute convergence of the sums Sk := ∑ n≥1, kn<k r−k n , for k = 1, 2, 3, . . . . The logarithmic derivative of (1) is f ′(z) f(z) = d dz log f(z) = d dz g(z) +m log z +∑ n≥1 ( log ( 1− z rn ) + z rn + 1 2 ( z rn )2 + · · ·+ 1 kn ( z rn )kn) = g′(z) + m z + ∑ n≥1 − 1 rn  1 1− ( z rn ) + 1 rn + z r2 n + · · ·+ z kn−1 rn n  = g′(z) + m z + ∑ n≥1 [ − ( 1 rn + z r2 n + z r3 n + · · · ) + 1 rn + z r2 n + · · ·+ z kn−1 rn n ] = g′(z) + m z − ∑ n≥1 [ zn rn n + zn rn n + · · · ]