Dichteschranken für die Lösbarkeit gewisser linearer Gleichungen.
In this paper, we study differential equations arising from the generating functions of the generalized Bell polynomials.We give explicit identities for the generalized Bell polynomials. Finally, we investigate the zeros of the generalized Bell polynomials by using numerical simulations.
Let be the -th Fibonacci number. Put . We prove that the following inequalities hold for any real :1) ,2) ,3) .These results are the best possible.
We determine decomposition properties of Euler polynomials and using a strong result relating polynomial decomposition and diophantine equations in two separated variables, we characterize those g(x) ∈ ℚ [x] for which the diophantine equation with k ≥ 7 may have infinitely many integer solutions. Apart from the exceptional cases we list explicitly, the equation has only finitely many integer solutions.
We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called “quotient problem” (resp. "-th root problem"), which in short asks whether the integrality of the values of the quotient (resp. -th root) of two (resp. one) linear recurrences implies that this quotient (resp. -th root) is itself a recurrence. We shall also relate such...
In this paper, we study triples and of distinct positive integers such that and are all three members of the same binary recurrence sequence.