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It is well known that the operator D = d/dz maps Z[z] into itself. But for Dn we have even more, namely (Dn/n!): Z[z] → Z[z]. Such effects are usually called “reduction of factorials”. They play an important role in studying arithmetic properties of various objects, especially values of hypergeometric functions. The author studies this property in a situation which is as general as possible. Let {fn,j}n∈N,j∈J be a family of algebraic numbers. We say that it satisfies the condition of “reduction of factorials ” if for some sequence {ψn ∈ Z>0}n∈N there holds: fk,jψn are algebraic integers for j ∈ J, 0 ≤ k ≤ n, n = 0, 1,... and lim sup ψ 1/n n ≤ ψ < ∞. Numbers fn,j arise usually as values of polynomials or Taylor coefficients of functions. Most of the author’s statements are based on the properties of binomial polynomials ∆n(z) = z(z + 1) · · · (z + n − 1)/n!, but he also studies ∆n(D), where D may be an algebraic number or the above operator or a matrix, and shows its connections with differential equations. In many situations, the effect of reduction of factorials was mainly established earlier, but the author’s approach gives explicit construction

Year: 2013

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