The distribution of zeros of generalq-polynomials

Abstract

Abstract.A general system of q-orthogonal polynomials is defined by means of its three-term recurrence relation. This system encompasses many of the known families of q-polynomials, among them the q-analogue of the classical orthogonal polynomials. The asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of the coefficients of the recurrence relation. This result is applied to specific classes of polynomials known by the names q-Hahn, q-Kravchuk, q-Racah, q-Askey and Wilson, Al Salam-Carlitz and the celebrated little and big q-Jacobi.

Zero Distribution of Orthogonal Polynomials on a q-Lattice

“…for every continuous function f on [0, 1]. This asymptotic distribution was also found in [1] using the moments of the asymptotic zero distribution and the asymptotic behavior of the coefficients in the three-term recurrence relation. In order to find a more interesting distribution of the zeros, i.e., a limit distribution which is not degenerate at 0, we need some scaling of the zeros.…”

Distributions of zeros of discrete and continuous polynomials from their recurrence relation

“…This method, which will be described in Section 2, is of general vality since no peculiar constraints are imposed upon the coe cients of the recurrence relation. It was found in a context of tridiagonal matrices 6, 13, 14, 15] and it has been already used for the study of the distribution of zeros of q-polynomials 1,11,16]. Some of the results found here have been previously obtained by other means and are dispersely published, what will be mentioned in the appropiate place; they are included here for completeness, for illustrating the goodness of our procedure or because they are not accessible for the general reader 16].…”

Μελέτη των ριζών των associated ορθογωνίων q-πολυωνύμων

“…Sth fusik paÐzoun shmantikì rìlo sth gwniak orm (angular momentum) kai sto q-anlogì thc, sthn q-Schrödinger exÐswsh kai stouc q-armonikoÔc talantwtèc (bl. [2] kai tic anaforèc pou uprqoun ekeÐ). Epiplèon ta orjog¸nia q-polu¸numa sundèontai me touc suntelestèc Glebsch-Gordan (3j kai 6j sÔmbola) oi opoÐoi èqoun shmantikèc efarmogèc sth fusik .…”

“…kai en α(r; c|q) = cq +ε(r; c|q) eÐnai to aristerì kro tou antÐstoiqou diast matoc orjogwniìthtac, tìte 2 . Lambnontac up' ìyin ìti toã n (r) = a n (r) fjÐnei wc proc r kai ìti tob n (r) = −b n (r) den auxnei wc proc r, sumperaÐnoume ìti h megalÔterh idiotim x n,1 (r) tou telest T n (r) me stoiqeÐaã j (r),b j (r) prèpei na fjÐnei wc proc r. Opìte, h mikrìterh rÐza x n,n (r) = −x n,1 (r) twn associated Legendre poluwnÔmwn auxnei wc proc r. Apì ta prohgoÔmena èpetai ìti to aristerì kro α n (r) tou diast matoc orjogwniìthtac eÐnai mh fjÐnousa sunrthsh tou r kai efìson α n (0) = −1 kai α n (r) ≤ −1, prèpei na eÐnai α n (r) = −1 gia r > 0.…”

“…Kpoia apotelèsmata sqetik me th monotonÐa twn riz¸n twn poluwnÔmwn Al-Salam-Carlitz dÐnontai sthn [65] kai gia tic rÐzec twn continuous q-Jacobi poluwnÔmwn sthn [9]. 'Oson afor sthn katanom twn riz¸n touc, sthn [2] melettai h diakrit puknìthta twn riz¸n (dhlad o arijmìc twn riz¸n an monda diast matoc) ka-j¸c kai to asumptwtikì thc ìrio, miac oikogèneiac orjogwnÐwn q-poluwnÔmwn h opoÐa perilambnei wc eidikèc peript¸seic ta perissìtera apì ta gnwst orjog¸-nia q-polu¸numa. Sthn paroÔsa diatrib xekin¸ntac apì thn anadromik sqèsh pou ikanopoioÔn ta associated orjog¸nia q-polu¸numa kai qrhsi-mopoi¸ntac mia sunarthsiak analutik mèjodo, dÐnontai apotelèsmata sqetik me th monotonÐa, thn kurtìthta kai ta ajroÐsmata Newton twn riz¸n orismènwn oikogenei¸n associated orjogwnÐwn q-poluwnÔmwn.…”