Alternative Orthogonal Polynomials. Vladimir Chelyshkov

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1 Aterative Orthogoa oyomias Vadimir Cheyshov Istitute of Hydromechaics of the NAS Uraie Georgia Souther Uiversity USA Abstract. The doube-directio orthogoaizatio agorithm is appied to costruct sequeces of poyomias which are orthogoa over the iterva [] with the weightig fuctio. Fuctioa ad recurret reatios are derived for the sequeces that are the resut of the iverse orthogoaizatio procedure. Quadratures geerated by the sequeces are described. Famiy of commo orthogoa poyomias origiates from a probem o the differetia equatio of the hypergeometric type which soutio is subected to certai additioa requiremets []. The poyomias aso may be defied by a orthogoaizatio procedure if it is appied to the fudameta sequece { x } i the order of power icrease. Geeraizig this approach oe ca deveop the orthogoaizatio procedure begiig with a arbitrary umber of the sequece both i the direct ad iverse order. The doube-directio agorithm of orthogoaizatio was itroduced i [2] for defiig orthogoa sequeces of expoets. It was aso metioed there that the agorithm may be appied to the fudameta sequece uder various orthogoaity reatios ad for the poyomias costructed the iverse agorithm retais the properties of the origia sequece as x if x []. Here we describe a exampe of such aterative sequeces. The aterative orthogoa poyomias (AO) obtaied are ot soutios of the equatio of the hypergeometric type but they ca be expressed i terms of the Jacobi poyomias. Let be a fixed whoe umber ad are sequeces of poyomias = { } = = τ x () = = { } = Tx (2) that hod orthogoaity reatioships dx = /( + + ) = =... (3)

2 ad ormaizig coditios dx= = +... (4) /( + + ) = sig( τ ) = ( ) sig( T ) = ( ). (5) The coefficiets τ ad T of the poyomias ad are defied uiquey by requiremets (3) - (5) ad the Gram-Schmidt orthogoaizatio agorithm which is reaized i the order of decreasig from to for sequeces () ad i the order of icreasig origiatig from = for sequeces (2). The sequeces ad have differet properties if x. For fixed ad x ~ x =... ad ~ x = The sequece represets the shifted to the iterva [] Legedre poyomias; > is cosidered as a auxiiary sequece ad the sequece is itroduced here as the aterative Legedre poyomias (AL). The poyomias ad have properties which are aaogues to the properties of commo orthogoa poyomias. Sice is fixed the poyomias ca be ( immediatey coected to a fixed set of the Jacobi poyomias αβ) m ( ξ )[] by verifyig property (4) ad (5) ad the foowig represetatio hods x = x x (6) (2 ) ( ) ( 2 ). This reatio ca be used directy to describe the properties of ad oe of the formuas that foow from (6) is the itegra represetatio + z ( z) = dz. (7) + 2 πi x ( z x) C Here C is a cosed curve which ecoses poit z = x. Reaizig the orthogoaizatio procedure oe ca suppose that the expicit defiitio of the poyomias is = ( ) x =.... (8) = This yieds Rodrigues type represetatio d + + = ( x ( x) ) =... (9) + ( )! x dx

3 ad orthogoaity reatioship (3) is cofirmed by appyig ast formua. It aso foows from (9) that ( xdx ) = xdx=. () + Maig use of formua (9) ad the itegra formua Cauchy for derivatives of a aaytic fuctio oe ca obtai the itegra represetatio ( + + 2) z ( z) x = πi x + ( z x ) C ( ) dz () wherec is a cosed curve which ecoses poit ead directy to the reciprocity reatio z = x. Represetatios (7) ad () = x ( x ). (2) ( + ) ( + ) Reatioship simiar to (2) hods aso for orthogoa expoetia poyomias [2]. Formua (2) faciitates descriptio of the AL ad the resuts that are show beow ca be obtaied maig use of the auxiiary sequeces ( x ) ad reatioship (2). I particuar = x = x + x 2 (2 ) ad the foowig recurrece reatios ad differetiatio formuas hod: a = ( b x c ) d (3) + α x( x) = ( β γ x) δ x + (4) κ xx ( ) = ( λ µ x) ν x (5) where a = ( + )( + )( + + ) b = (2+ )(2+ 2) c = (2 + )(( + ) + + ) d ( )( 2) = + + α = 2( + ) β = 2 ( + ) γ = + + 2

4 δ = ( )( + + 2) κ = 2 λ = 2 ( + ) µ = + + ( ) ν = ( + )( + + ). (6) The poyomia x is a soutio of the differetia equatio ζ ζ ζ = = (7) 2 x x (( ) x ( ))... that aso foows from the costructios deveoped. Maig use of the substitutio ς = xux ( ) oe ca represet the poyomia soutio of equatio (7) i terms of the hypergeometric fuctio F ad the foowig reatioships hod + = xf ( + + 2;2+ ; x) (2 ) = x ( 2 x) =.... Thus the AL are reated to differet famiies of the Jacoby poyomias. Let f C[]. The AL x as x ad they ca be appied for costructig the Gauss-type quadrature formua. Maig use of the orthogoaity reatios ad the approach deveoped i [3 page 378] oe ca obtai the formuas f ( xdx ) wf (8) = where x are the zeros of the poyomia = 2... ad the weightig factors w are w = 2 (2+ ) ( x ). (9) Aterative Gauss quadrature (AGQ) (8) (9) is exact for x 2 2. Aaogues approach was described for orthogoa expoetia poyomias i [4]. Taig ito accout property () of the AL oe ca come to the cocusio that the Radautype quadrature is ecessary to add the fuctio x 2 2 to the above oes exacty itegrabe. I particuar if = such a quadrature is correct for the fuctio f. To avoid appicatio of the aterative Radau quadrature amost orthogoa sequece of

5 poyomias { } = couped with the AGQ might be cosidered for probems o approximatio. Ay poyomia fuctio say m x [] ca be cosidered as a term of the aterative poyomia hierarchy A = { Q} = Q = { Q} = Q = x where is the spa of the poyomia Q ( x ). Beig subected to certai orthogoaity reatios aterative poyomias eep distictivey a the attributes of reguar orthogoa poyomias. Thus pecuiarity of such a costructio has ed to descriptio of the sequece of orthogoa expoetia poyomias that geerate Gaussia quadrature for expoets [2] [4]. The agorithm of iverse orthogoaizatio of the fudameta sequece resuts i redistributio of the zeros of commo orthogoa poyomias that maes possibe deveopig differet Gauss-type quadratures. This faciitates appicatio of the AO to itegratig the iitia vaue probem [5]. Refereces. Niifirov A.F. Uvarov 978. V.B. Specia fuctios of mathematica physics. Naua Moscow 33p. (i Russia) 2. Cheyshov V.S Sequeces of expoetia poyomias which are orthogoa o the semiaxis. Reports of the Academy of Scieces of the Uraie (Doady AN USSR). ser. A N : pp. 4-7 (i Russia). 3. Laczos C. Appied Aaysis. retice Ha Egewood Ciffs 539p. 4. Cheyshov V.S A variat of spectra method i the theory of hydrodyamic stabiity. Hydromechaics (Gidromehaia). N 68: pp. 5-9 (i Russia). 5. Cheyshov V. S. 2. A spectra method for the Cauchy robem Sovig. roceedigs of MCME Iteratioa Coferece. robems of Moder Appied Mathematics. WSES ress pp