Y. Rouba, K. Smatrytski

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1 Geeratig fuctio ad its applicatio for geeralizatio of Legedre polyomials РАЗНОЕ УДК 53.5 Y. Rouba, K. Smatrytsi GENERATING FUNCTION AND ITS APPLICATION FOR GENERALIZATION OF LEGENDRE POLYNOMIALS 49 Проводится обобщение полиномов Лежандра с помощью производящей 49 функции. Найден явный вид искомых функций. Рассмотрены неко- торые частные случаи. Особое внимание уделено случаю, когда параметры, определяющие изучаемые функции, являются различными, симметричными относительно нуля действительными числами. Изучены некоторые свойства этих функций. Основываясь на результатах численных экспериментов, выдвинута гипотеза о корнях исследуемых функций. I the preset paper the geeralizatio of Legedre polyomials with the help of geeratig fuctio is studied. The eplicit form of cosidered fuctios is foud. Some special cases are cosidered. Particular attetio is paid to the case whe the parameters defiig the studied fuctios are differet, symmetric about zero real umbers. Some properties of costructed fuctios are obtaied. Based o the results of umerical eperimet a hypothesis about zeroes of these fuctios is stated. Ключевые слова: полиномы Лежандра, производящая функция, ортогональные системы. Key words: Legedre polyomials, geeratig fuctio, orthogoal systems. Itroductio Geeratig fuctio is oe of the classical ways for costructio of orthogoal systems. Applicatio of geeratig fuctios for costructio of orthogoal systems of algebraic polyomials is described i detail i the []. I the case of ratioal fuctios this problem is more complicated. I 964 M. M. Dzhrbashya ad A. A. Kitbalya applied geeratig fuctio for costructio of systems of ratioal fuctios which geeralized Chebyshev polyomials of the first ad the secod type []. It should be metioed that i the wor [3] the costructio of system of ratioal fuctios which geeralized Jacobi polyomials with respect to the weight ( )/( ) was described. I the preset paper the geeralizatio of Legedre polyomials with the help of geeratig fuctio is cosidered. It s well ow that the fuctios Fz (, ) z z, (,), F (,0), Rouba Y., Smatrytsi K., 04 Вестник Балтийского федерального университета им. И. Канта. 04. Вып. 0. С

2 Y. Rouba, K. Smatrytsi is a geeratig fuctio for algebraic Legedre polyomials, i. e. P ( ) Fz (, ) z, z. 0! I other words, the Legedre polyomials are the Taylor coefficiets of the fuctio Fz (, ), dz P ( ) F(, z) i z 50 For the geeralizatio of Legedre polyomials o ratioal case we cosider 50 the followig ratioal fuctios. Let the sequece { } 0 of comple umbers be such that 0 0,, 0. Usig this umbers we defie the fuctios: z. Mai result, 0. ( z ) 0, z ( z) z z, 0. Lemma []. The system { ( z )} 0 is a orthogoal o the uit circle z. Note, that if z the z z ( z). () z 0 z z z Here ad later the fuctio ( z ) is defied by formula () also i the case z. The fuctios, which geeralize the Legedre polyomials, we defie as follows dz L( ) F(, z) ( z) i z, () where is a circle z, 0, such that,,,...,. Remar. If 0,,,...,, the () z z, ad dz L () F(,) z, i z z i. e. i this case L( ) is a Legedre polyomials, orthogoal o the segmet [,] with respect to the weight. Theorem. If 0,,,...,, ad,,,,,...,, the the followig formula holds 0 ( ) L( ) ( ) (3)

3 Geeratig fuctio ad its applicatio for geeralizatio of Legedre polyomials Proof. Usig formula () we obtai i z z z z z z dz L( ). The we apply the substitutio ( t ) z z zt or z. (4) t I this case t t t t dz dt, z z, ( ) t t 5 5 z t ( t) ( t ) ( t), z. z ( t ) ( t) t Therefore t t ( t) i C t t t t t L( ) dt, ( ) ( ) ( ) ( ) where C is a image of whe mappig tz ( ). Uder the coditios of the theorem the itegrad t t ( t) Ft () t ( t ) ( t ) ( t ) ( t ) has simple poles at the poit t ad at the roots of the equatios ( t ) ( t ),,,...,, i. e. t,, t,,,,...,. It is ot difficult to show that the poits t,,,,...,, are iside the curve C, ad the poits t,,,,...,, are outside this curve. To prove it oe eeds to fid the correspodig poits z, z( t, ),,,...,,, usig (4) ad verify that z, z,, z,,,,...,. By the Cauchy s residue theorem we get L( ) Res( F, ) Res( F, t,). (5) It is easy to see that ( ) Res( F, ). (6) Now we calculate residues at the poits t,,,,...,. We have t ( t) t ( t) Res( Ft, ). t, t ( t ) ( t) ( t ) ( t ) ( t t,) tt, tt, tt tt,,

4 Y. Rouba, K. Smatrytsi The we fid t, t tt, ( t ) ( t) t tt, ( t) t, 5 5 t t ( ) t t, ( ) ( ) t t t tt, t tt, t ( t ) t ( t) t ( tt ) ( tt ), tt,, tt, tt, ( ). Sice that, Res( Ft,,). (7) ( ) Ad the residue at the poit t, t t ( t) Res( Ft,,) t ( tt,) ( ) ( ) t t t t, tt, tt, (8). To complete the proof oe must put (6), (7) ad (8) ito (5). Corollary. By the formula (3) the fuctio L( ) ca be writte i the form c L( ) 0, where c,,,...,, are some comple umbers. Its sigular poits are as follows,,,...,.

5 Geeratig fuctio ad its applicatio for geeralizatio of Legedre polyomials. Some importat cases ) Obviously, L ( ). 0 ) Let 0 0,,. The the secod summad i the right side of (3) vaishes, ad the product i the third summad is equal to. Sice that, Note that L( ). 53 lim L 53 ( ) lim lim Besides, the fuctio L ( ) has the oly zero at the poit. 3) Let the sequece { } 0 be as follows: 0 0,,,..., (distict poits), 0. I this case z z ( z), ( z) 0 z z z ad i a similar way t t ( t) L( ) dt 4 i ( t) ( t ) ( t ) C or - L( ) Res(, ) Res(,,) F F t, where t t ( t) Ft (), ( t ) ( t ) ( t) t, is the root of the equatio ( ) ( ) 0 which lies iside the t t curve C. Now we fid residues. The poit t is a pole of the secod order. Thus t ( t ) Res(F,) ( t ) ( t ) ( t) t ( ) t ( t) ( ) ( ) ( t ) ( t) t ( ) ( ) ( ) ( ). ( ) ( )

6 Y. Rouba, K. Smatrytsi The poits t,,,,...,, are the simple poles. We have Res( Ft, ) t t ( t) t ( t) ( t) ( tt ) ( t ) ( t),, ( ) Fially, ( ) L( ). (9) tt, ( ) Now we cosider the followig sequece of parameters. Let,,..., be a sequece of distict real positive umbers. The the parameters { } 0 are as follows: 0 0,,,,,...,, 0. I this case by the formula (9) we get L( ). These fuctios are odd, L (). Besides usig umerical eperimet we ca draw the graph of some of these fuctios. I the figures,, 3 we have the graphs of L 3 ( ), L ( ), L ( ) for , 0.6, Fig.. The graph of L ( ) for Fig.. The graph of L ( ) for Fig. 3. The graph of L ( ) for

7 Geeratig fuctio ad its applicatio for geeralizatio of Legedre polyomials Cocerig these results some questios ca be raised: (a) Does fuctio L() has distict real zeroes i the iterval (, )? (b) Numerical results show that the system { L( )} is ot orthogoal. Is there a sequece { } such that correspodig system { L( )} is orthogoal? Note, that if the aswer to the first questio is positive tha we ca use these zeroes as odes for costructio of iterpolatio polyomial. Refereces. Sueti P. K. Classical orthogoal polyomials. Mosow, 978. (i Russia) 55. Dzhrbashya M. M, Kitbalya A. A. O oe geeralizatio of Chebyshev polyomials // Reports of Armeia SSR Academy of sciece Vol. 38, N P (i Russia) 3. Rovba E. A. Orthogoal systems of ratioal fuctios o the segmet ad quadratures of Gauss-type // Mathematica Balaica Vol. 3, N. P Об авторах Евгений Алексеевич Ровба д-р физ.-мат. наук, проф., Гродненский государственный университет им. Янки Купалы, Беларусь. rovba@grsu.by Константин Анатольевич Смотрицкий канд. физ.-мат. наук, доц., Гродненский государственный университет им. Янки Купалы, Беларусь. .smotritsi@grsu.by About the authors Prof. Yauhei Rouba Yaa Kupala State Uiversity of Grodo, Belarus. rovba@grsu.by Dr Kastati Smatrytsi Ass. Prof., Yaa Kupala State Uiversity of Grodo, Belarus. .smotritsi@grsu.by