On Polynomials Construction

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1 Intenational Jounal of Mathematical Analysis Vol., 08, no. 6, 5-57 HIKARI Ltd, On Polynomials Constuction E. O. Adeyefa Depatment of Mathematics, Fedeal Univesity Oye-Ekiti P.M.B. 373, Oye-Ekiti, Nigeia Coesponding autho Y. Hauna Depatment of Mathematics, Sa adatu Rimi College of Education Kumbotso, Kano, Nigeia R. O. Aewole Depatment of Mathematics, Fedeal Univesity Oye-Ekiti P.M.B. 373, Oye-Ekiti, Nigeia R. I. Ndu Depatment of Mathematics, Rives State Univesity, Rives State, Nigeia Copyight 08 E. O. Adeyefa, Y. Hauna, R. O. Aewole and R. I. Ndu. This aticle is distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. Abstact The solutions of many physical poblems ae obtained as expansions in seies of othogonal functions. In this pape, constuction of new class of othogonal polynomials is poposed. This class of polynomials is valid in the inteval [-, ] with espect to constant factional weight function, w. The ecuence elation of the class of polynomials is pesented as we hope to futhe discuss additional popeties and full application of the class of polynomials in the futue pape. Keywods: Othogonal polynomials, Weight function

2 5 E. O. Adeyefa et al.. Intoduction Polynomials ae impotant tools in appoximating a function. The choice of polynomial depends on the behavio and type of such polynomials. Among the well-known polynomials ae Chebyshev, Legende, Hemite and Laguee polynomials. In this wok, we popose fomulation of a new class of othogonal polynomials valid in inteval [-, ] with espect to constant weight function, w.. Methodology Hee, we want to deive the new othogonal polynomials ove the inteval [-,] with espect to weight function w(x) =. The pocedue euies choosing the othogonal polynomial (x) defined as n (x) = C (n) x () =0 whee C (n) ae the othogonal coefficients. Hence, the following euiements ae consideed m (x), n (x) = w(x) m (x) n (x) dx = h n δ mn () with δ mn = { 0 m n m = n We adopt a popety called the nomalization given as Fo = 0 in (), we have n () = (3) Fom (3), we have 0 (x) = C 0 (0) 0 (x) = C 0 (0) = Hence, 0 (x) = (4)

3 On polynomials constuction 53 Fo = in (), we have Fom (3), we obtain (x) = C 0 () + C () x (5) Applying the inne poduct popety in (), we have C 0 () + C () = (6) 0 (x), (x) = 0(x) (x) dx = 0 which implies C 0 () = 0 (7) On solving (6) and (7) and then substituting the outcomes into (5), we have (x) = x (8) When = in (), we have (x) = C () 0 + C () x + C () 3 x (9) By definition (3), euation (9) gives C () 0 + C () + C () 3 = (0) and 0 (x), (x) = 0(x) (x) dx = 0 This implies, 3 C () + C () 0 = 0 () Also, (x), (x) = (x) (x) dx = 0 which leads to

4 54 E. O. Adeyefa et al. 3 C () = 0 () Solving (0), () and () and substituting the outcomes into (9), we have (x) = 3 x (3) Let = 3 in (), we have Using (3), euation (4) give 3 (x) = C 0 (3) + C (3) x + C 3 (3) x + C 3 (3) x 3 (4) C 0 (3) + C (3) + C 3 (3) + C 3 (3) = (5) Consideing the inne poduct 0 (x), 3 (x) = 0(x) 3 (x) dx = 0 which on simplification gives 3 C (3) + C (3) 0 = 0 (6) Consideing the second inne poduct, we have (x), 3 (x) = (x) 3 (x) dx = 0 Which is futhe simplified to have 5 C (3) + 3 C (3) = 0 (7) Likewise, (x), 3 (x) = (x) 3 (x) dx = 0 which on simplification gives 5 C (3) = 0 (8)

5 On polynomials constuction 55 Solving(5), (6), (7), (8) and substituting the outcome into (4), we have 3 (x) = 5 x3 3 x (9) In the same manne, 4 (x) to 0 (x) ae geneated and heeunde ae the polynomials. 0 x 3 x x - x x - x x - x x x - x x x - x x x x - x x x x - x x x x x - x x x x (0) In the spiit of Fische and Golub, euation (3) must satisfy thee-tem ecuence elation c p ( t a ) p p b p,,,..., p ( t) 0, p0 0 whee b, c> 0 fo (b is abitay).

6 56 E. O. Adeyefa et al. Thus, c p ( n ) Pn ( x), ( t a ) p (n ) xpn, b p npn, n,,... Hence, the ecuence elation fo this othogonal polynomial is theefoe given as P x) n (n ) xp np, n, P, P ( x x n ( n n 0 ) The euation above, togethe with P0 and P x allows the new set of polynomials to be geneated ecusively. 3. Test of Applicability We investigate hee, the applicability of this class of othogonal polynomials by consideing the popula fou-step Adams-Moulton explicit method. Fo this pupose, sk 0 y a () and sk 0 y ' a ' () Intepolating () at x = xn+3 and collocation () at x = xn+k, k=0()4, we obtain a system of six euations which ae solved simultaneously and the esulting values of a ae substituted back into (0) to have the continuous schemes. Evaluating the continuous scheme at x = xn+4 yields y y h (5f f 64 f 06 f 9 f n 4 n3 n4 n3 n n n which is the popula fou-step Adams-Moulton explicit method. ) 4. Conclusion A new class of othogonal polynomials with ecuence elation has been geneated in this wok. This class of new polynomials, when applied, ecoves the well-known fou-step Adams-Moulton method.

7 On polynomials constuction 57 We theefoe ecommend this class of new polynomials fo geneal puposed use while futhe investigation is ongoing on the othe basic popeties as we hope to discuss this in the futue pape. Refeences [] B. Fische and G.H. Golub, On geneating polynomials which ae othogonal ove seveal intevals, Math. Comp., 56 (99), Received: Apil 3, 08; Published: May 3, 08

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