Formal power series of logarithmic type
Synopsis of this 118 pages long paper (or better booklet) written by the authors has more than 14 pages, and shows the difficult role of the reviewer. A potential reader of this very lucidly written paper can perhaps obtain the first information about this work from these extracts from the authors' introduction: Since its beginning with Barrow and Newton, the calculus of finite differences has been viewed - whether admittedly or not - as a formal method of computation with special functions. Until nineteenth century, when the rigors of convergence were to be gradually imposed, mathematicians handling difference equations and series involving polynomials and logarithms, were not beset by doubts as to the correctness of their manipulations, and in fact, their results have seldom turned out to be incorrect, even by the standards of our day.... What is needed in order to justify formal manipulations - we can state nowadays with the confidence that comes after fifty years of local algebra - is an extension of the notion of formal power series that will include, besides powers of x, power series in other special functions of classical analysis, most notably exponentials and logarithms. It is the purpose of this work to carry out such an extension, one that includes the formal theory of infinite series in polynomials and logarithms. The difficulty of this program, and perhaps the reason why it has not been previously carried out, is the algebraic unwildiness of the functions \(x^ n(\log x)^ t\), where n is an integer and t a nonnegative integer. The coefficients in the Taylor expansions in power of a of the function \((x+a)(\log (x+a))^ t\) do not seem to follow rhyme or reason, and unless these coefficients are somehow made easy to handle, no simple operational calculus can be obtained. We resolve this difficulty by introducing another basis for the vector space spanned by the function \(x^ n(\log x)^ t\), a basis whose members we call the harmonic logarithms. We denote by \(\lambda_ n^{(t)}(x)\) the harmonic logarithm of order t and degree n. For positive integers t and integers n, or for \(t=0\) and nonnegative integers n, the harmonic logarithm of order t and n span the subspace \(L^{(t)}\) of the logarithmic algebra spanned by \(x^ n(\log x)^ t\) over the same values of n and t. In particular, \(L^{(0)}\) is the subspace of ordinary polynomials in x. When t is a positive integer, however, the variable n is allowed to vary over all integers, positive or negative. One has, for instance, for \(t=1\), \[ \lambda_ n^{(1)}=x^ n(\log x-1--...-1/n)\quad for\quad n\geq 0,\quad =x^ n\quad for\quad n<0. \] The harmonic logarithm turn out to satisfy an identity which ``logarithmically'' generalizes the binomial theorem for polynomials, to wit: \[ \lambda_ n^{(t)}(x+a)=\sum_{k\geq 0}\lfloor \begin{matrix} n\\ k\end{matrix} \rceil a^ k\lambda^{(t)}_{n-k}(x). \] The coefficients \(\lfloor \begin{matrix} n\\ k\end{matrix} \rceil\) are generalizations of the binomial coefficients (and, in fact, coincide with the binomial coefficients in all cases in which the binomial coefficients are defined). We propose to call them the Roman coefficients. They satisfy identities similar to those satisfied by the binomial coefficients, and they are defined for all integers n and k. The ring of formal series of logarithmic type is not defined in two steps in terms of the harmonic logarithms. First, one completes the subspace \(L^{(t)}\) into a vector space \({\mathcal L}^{(t)}\) in such a way as to obtain formal series of the form \[ (1)\quad \sum^{n\leq d}b_ n\lambda_ n^{(t)}(x), \] and second, one takes the algebraic direct sum \({\mathcal L}\) of the vector spaces \({\mathcal L}^{(t)}\), thereby obtaining the logarithmic algebra. Thus, a formal series of logarithmic type is a finite sum of (infinite) formal series of the form (1) for different values of t. Note that for \(t=0\), one obtains nothing more than polynomials in x. The ring of formal differential operators with constant coefficient acts on the logarithmic algebra \({\mathcal L}\). More pleasingly, the ring of all formal Laurent series in the derivative D acts on the subspace \({\mathcal L}^{(+)}\) which is the direct sum of all subspaces \({\mathcal L}^{(t)}\) for t positive. The subspace \({\mathcal L}^{(+)}\) turns out to be the ``largest'' subspace of the logarithmic algebra on which the derivative operator is invertible. This fact (together with certain commutation relations explained in the text) leads to a definition of natural solutions of difference equations which is more general than the one previously given by Milne-Thompson and Nörlund, and which coincides with terms in all cases where both are defined. The rest of this work develops logarithmic analogs of various notions that were previously only known for polynomials, notably the logarithmic analog of Appel polynomials and the logarithmic analog of the theory of sequences of binomial type, which we call Roman graded sequences. Several special cases are worked out, notably the logarithmic analogs of Bernoulli and Hermite polynomials, of Gold and Laguerre polynomials, and of factorial powers of the calculus of finite differences. In the last example, one finds that the Gauss \(\psi\)-function (the logarithmic derivative of the gamma function) is one term in the logarithmic extension of the lower factorial function. Thus, classical identities satisfied by the \(\psi\)- function are seen to be trivial consequences of general logarithmic identities. We stress that the examples given here are only a sampling of the special functions that can be brought under the logarithmic umbrella.
scientific article; zbMATH DE number 4127206
scientific article