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Some recurrence formulas related to the differential operator $$\theta$$D. (English) Zbl 1034.11022
The author’s primary concern is the differential operator $$\theta$$D, where D is the derivative symbol and $$\theta$$ is an element of a differential field. Properties of this operator are used to derive several identities, including the classical expression for the sum $$1^k+2^k+\cdots+(n-1)^k$$, where $$k\geq2$$ is a given natural number. The paper is concluded by a discussion of the algebraic theory of differential operators. As an example, the authors consider Kurepa’s function (see e.g., the paper by the first author and the reviewer in [Publ. Inst. Math., Nouv. Sér. 57(71), 19–28 (1995; Zbl 0873.11015)]) $K(z)=\int_0^\infty e^{-t}\frac{t^z-1}{t-1}\,dt \quad (\operatorname{Re}z>0).$ It is proved that $$K(z)$$ is differentiably transcendental over the differential field of real, rational functions. The proof follows from the corresponding result for the gamma-function, since $$K(x+1)-K(x)=\Gamma(x)$$.

MSC:
 11B75 Other combinatorial number theory 11B73 Bell and Stirling numbers 12H05 Differential algebra