Oberschelp, Walter Solving linear recurrences from differential equations in the exponential manner and vice versa. (English) Zbl 0862.11013 Bergum, G. E. (ed.) et al., Applications of Fibonacci numbers. Volume 6: Proceedings of the sixth international research conference on Fibonacci numbers and their applications, Washington State University, Pullman, WA, USA, July 18-22, 1994. Dordrecht: Kluwer Academic Publishers. 365-380 (1996). The author investigates connections between differential equations and sequences of numbers satisfying recurrence relations. Among others the following theorem is proved: “Let \(f(z)= \sum f_n (z^n/n!)\) be holomorphic at the origin. Then \(f (x)\) fulfills a linear differential equation \[ p_r(z) f^{(r)} + p_{r-1} (z)f^{(r-1)} + \cdots + p_1(z) f' + p_0(z)=a(z) \] iff \(\{f_n\}\) fulfills a linear difference equation via the following symbolic term bijection: \[ {z^k \over k!} f^{(s)} \Leftrightarrow {n\choose k} f_{n+s-k}. \text{''} \] Using the results, generating functions and explicit forms of the terms can be obtained for recurrence sequences.For the entire collection see [Zbl 0836.00026]. Reviewer: Péter Kiss (Eger) MSC: 11B37 Recurrences 34A30 Linear ordinary differential equations and systems Keywords:differential equations; recurrence relations; difference equation; recurrence sequences PDFBibTeX XMLCite \textit{W. Oberschelp}, in: Applications of Fibonacci numbers. Volume 6: Proceedings of the sixth international research conference on Fibonacci numbers and their applications, Washington State University, Pullman, WA, USA, July 18-22, 1994. Dordrecht: Kluwer Academic Publishers. 365--380 (1996; Zbl 0862.11013)