Newton sum rules and monotonicity properties of the zeros of scaled co-recursive associated polynomials

Abstract

ABSTRACT. Let Q n (a;;/3,7,c) be the polynomials of degree n which satisfy the recurrence relation: a!n+cQn+i(a;/3,7,c) + a! n+c _iQ n _i(z;/3,7,c) + (Pn+c + {3Sn,o)Qn(x; /?,7,c) = x(l + (7 -l)Sn,0)Qn(a; P, 7> c), Q_l(a;;/3,7,c) = 0, Qo(a;;^,7,c) = 1.In the above, P is real, 7 > 0, a n + c and /3n+c are real sequences with Q: n + C > 0, and 5 ni o is the Kronecker symbol. These polynomials are called scaled co-recursive associated polynomials. The co-recursive associated orthogonal polynomials are obtained from the above for 7=1. In this paper, the Newton sum rules for the k-th power of the zeros of scaled co-recursive associated orthogonal polynomials are determined in terms of the Newton sum rules of associated orthogonal polynomials. Some monotonicity properties of the zeros also are given.

Μελέτη των ριζών των associated ορθογωνίων q-πολυωνύμων

“…H mèjodoc pou ja qrhsimopoi soume gia th melèth thc monotonÐac kai thc kurtìthtac twn riz¸n twn associated q-poluwnÔmwn, eÐnai mia sunarthsiak analutik mèjodoc kai basÐzetai sthn anadromik sqèsh tri¸n ìrwn pou ikanopoioÔn aut ta orjog¸nia polu¸numa kai eis qjh apì touc Ifantis kai Siafarikas sthn [55] gia ta klasik orjog¸nia polu¸numa kai stic [56], [111] gia ta antÐstoiqa associated orjog¸nia polu¸numa. EpÐshc gia ton upologismì twn ajroismtwn Newton twn riz¸n twn associated orjogwnÐwn q-poluwnÔmwn h sunarthsiak analutik mèjodoc pou ja qrhsimopoihjeÐ parousisthke sthn [52] gia ton upologismì twn ajroismtwn Newton twn riz¸n twn scaled co-recursive associated orjogwnÐwn poluwnÔmwn.…”

“…, brÐskoume wc eidikèc peript¸seic, ta ajroÐsmata Newton twn riz¸n twn associated Laguerre, twn associated Charlier, twn associated Ultraspherical kai twn associated Meixner poluwnÔmwn pou èqoun brejeÐ prìsfata stic erga-sÐec [3,41,52,68,92,93].…”

“…AjroÐsmata Newton twn riz¸n twn associated orjogwnÐwn q-poluwnÔmwn EÐnai gnwstì apì th jewrÐa telest¸n ìti an M eÐnai ènac summetrikìc telest c se ènan pragmatikì q¸ro Hilbert peperasmènhc distashc, ìpwc o q¸roc H n , tìte to jroisma n j=1 M e j , e j eÐnai anexrthto apì thn orjokanonik bsh e j , j = 1, 2, ...n. 'Opwc kai sthn [52] eÐnai dunatìn na apodeiqjeÐ ìti, en x n,j (ν|q), j = 1, 2, ..., n, eÐnai to pl rec orjokanonikì sÔsthma idiodianusmtwn tou telest T (ν|q) ston H n , tìte n j=1 T k (ν|q)e j e j = n j=1 T k (ν|q)X n,j (ν|q), X n,j (ν|q) = x n,j (ν|q) = x k n,j (ν|q) gia opoiod pote k, oi upologismoÐ ìmwc pou apaitoÔntai eÐnai idiaÐtera polÔplokoi. a + 1 2 sto 0.…”

“…, brÐskoume ta ajroÐsmata Newton gia ta associated Ultraspherical polu¸numa , ta opoÐa eÐnai dunatìn na prokÔyoun kai wc eidikèc peript¸seic (a = b = λ − 1/2), apì tic antÐstoiqec sqèseic gia ta associated polu¸numa Jacobi oi opoÐec èqoun dojeÐ sta [52,92,93]. Wc ek toÔtou, ta ajroÐsmata Newton gia tic rÐzec twn associated continuous q-Legendre poluwnÔmwn prokÔptoun apì ta antÐstoiqa ajroÐsmata Newton gia tic rÐzec twn associated continuous q-Ultraspherical polu¸numa jètontac λ = q 1 2 .…”

“…Epiplèon, qrhsimopoi¸ntac diaforetikèc mejìdouc, o upologismìc twn rop¸n thc katanom c twn riz¸n isodÔnama twn ajroismtwn Newton twn riz¸n twn klasik¸n orjogwnÐwn poluwnÔmwn èqei gÐnei stic ergasÐec [3,15,16,20,40,41,68,85,91,104], twn hmiklasik¸n orjogwnÐwn poluwnÔmwn stic [90,124], twn associated kai twn co-recursive associated orjogwnÐwn poluwnÔmwn stic [68,92,109], twn scaled co-recursive associated orjogwnÐwn poluwnÔmwn stic [52,93], twn quasiorthogonal poluwnÔmwn thc klasik c kathgorÐac (classical class) sthn [126] kai twn relativistic poluwnÔmwn sthn [94].…”