B OLLETTINO U NIONE M ATEMATICA I TALIANA Richard Bellman A q-version of the Newton interpolation formula and some Eulerian identities. Bollettino dell’Unione Matematica Italiana, Serie 3, Vol. 16 (1961), n.3, p. 285–287. Zanichelli <http://www.bdim.eu/item?id=BUMI_1961_3_16_3_285_0> L’utilizzo e la stampa di questo documento digitale è consentito liberamente per motivi di ricerca e studio. Non è consentito l’utilizzo dello stesso per motivi commerciali. Tutte le copie di questo documento devono riportare questo avvertimento. Articolo digitalizzato nel quadro del programma bdim (Biblioteca Digitale Italiana di Matematica) SIMAI & UMI http://www.bdim.eu/ Bollettino dell’Unione Matematica Italiana, Zanichelli, 1961. A g-version of the Newton interpolation formula and some eulerian identifies Nota di RTCHARD BBLLMAN (*) Sammary. - Using an extension of the NEWTON interpolation formula, tt is shown how to obtain series expansions for fitnctions such as oo II (1 -h x qn). 1. Introduction. By means of the functional équation (1) f(x) = (1 +- xq)f(xq\ it is easy to dérive the identity 1 ' n } ~ n=o *;~ nti (i - g)(i — g9)..- (i - ?") ' This technique, introduced by EULER, ^vas extensively developed by GrAuss and JACOBI. In a recent paper, [1], Ave showed ho\\ r results of this nature could be obtained by means of partial fraction expansions. In this paper, we wish to present a third method for obtaining results such as (2), based upon an extension of Newton' s interpolation formula. 2. q-version of Newton Formula The classical interpolation formula of M. .. . „,» 1) NEWTOIST has the fornx n(n — l ) ( n - 2) where, as is easily established, (*) Prevenuta alla Segreteria dell'U. M.I. il 26/7/61. 286 RICHARD BELLMAN (2) aD = fl0), and so on. Let us now consider an extension of this of the form id for n = Q, 1, 2,..., To détermine the coefficients, we Write = a0 - Ol and subtract, obtaining An H- 1) - An) Hence, if we Write (6, we can Write 3. Application Write (1) fln) = A g-VERSION OF THE NEWTON INTERPOLATION FORMULA, ETC. 287 Usittg the foregoing results, we obtain the identity n (1 -+- xqk) = 1 Hh qx[ ~ -<- (2) Take | q | < 1 and let n -*- oo. We obtain as the limit of (2) the relation of (1.2). 4. Extensions In a similar fashion, we can dérive many other intèresting identities. Farthemore, the result can be extended to functions of several variables. We can write a. U.-1/Var-i / " i g,'-i J' ajid thereby dérive corresponding results for the function QO Finally; let us note, we can dérive further results by expanding (3) f{n) = n (1 ~h xqxh) ~ l o "where g does not necessarily equal g^. REFEÉBNOE [1] R. BELLMAN. Tfee expansions of some infinitéproducts, «Duke Math J.» , Vol. 24; 1957, pp. 353-3Ö6. 20