Formal power series of logarithmic type

Recent Contributions to the Calculus of Finite Differences: A Survey

“…More generally, the p α n (x) each belong to L α and in fact form a basis for it. 12 Obviously, for n nonnegative, ∆ n 1…”

Some Explanations of Dobinski's Formula

“…We present a geometric identity that was first derived from the study of mixed v.olumes of compact convex sets [5]. When the identity is considered in view of combinatorics, it admits an interesting generalization that is closely related to harmonic numbers and formal power series of logarithmic type [11].…”

“…It is interesting to notice that C(n, m) is related to harmonic numbers c~-n) [11]. In fact, if n is negative and m is positive, the harmonic number C~) can be represented by c~n) = E (_l)k-l(~)k-n. k=l (7) Then C(n, m) = -c~-n) whenever n < 0, m ~ 1.…”

“…Let nand m be two positive integers. The harmonic number C<;:) = -C( -n, m) can be represented combinatorially by [11] i.e., the harmonic number C<;:) is equal to the sum of reciprocal product of parts of linear partitions p with n + 1 parts and with no part greater than m.…”

“…The purpose in writing this note is to provide geometric, algebraic, and combinatorial methods to prove the identity (1). Let Bn be the nth Bell Moreover, H is interesting that the numbers C(n, m) (n> 0) have various combinatorial explanations and C(n, m) (n < 0) are the harmonic numbers that are related to formal power series of logarithmic type [11]. The employed method here may be a useful technique in proving some other combinatorial identities [2,7,16].…”