Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
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1 Ope Math. 2017; 15: Ope Mathematic Reearch Artice Hemut Prodiger* Repreetig derivative of Chebyhev poyomia by Chebyhev poyomia ad reated quetio Received Jue 26, 2017; accepted Augut 3, Abtract: A recurio formua for derivative of Chebyhev poyomia i repaced by a expicit formua. Simiar formuæ are derived for caed Fiboacci umber. Keyword: Chebyhev poyomia, Iverio formua, Expicit formua, Scaed Fiboacci umber MSC: 11B39 1 Itroductio Coider the Chebyhev poyomia of the ecod id U.x/. 1/.2x/ 0=2 2 I the mai iteret of the paper [4] i to repreet the derivative of U.x/ i term of the Chebyhev poyomia themeve. To thi aim a exact computatioa method (a recurio formua) wa preeted. I the preet ote, we repace thi computatioa method by a exact ad expicit formua. Our awer i U./.x/ / C 1. 2 C 1/U 2.x/: Athough it i ot eeded, we briefy metio without proof a aaogou formua for the Chebyhev poyomia of the firt id: Let T.x/. 1/ x 2 ; the 0=2 T./.x/ / C 1 T 2.x/. C 1 : ŒŒ eve 2.. C 1/ *Correpodig Author: Hemut Prodiger: epartmet of Mathematica Sciece, Mathematic iviio, Steeboch Uiverity, Private Bag 1, 7602 Matiead, South Africa, E-mai: hprodig@u.ac.za Ope Acce Prodiger, pubihed by e Gruyter Ope. Attributio-oCommercia-oeriv 4.0 Licee. Thi wor i iceed uder the Creative Commo
2 Repreetig derivative of Chebyhev poyomia157 We ue here the otio of faig factoria x W x.x 1/ : : :.x C 1/ ad Ivero ymbo ŒŒP which i if P i true ad 0 otherwie, compare [1]. I a at ectio, we tur our attetio to two other famiie of poyomia (caed Fiboacci umber). 2 The proof Our tartig poit i the iverio formua (ee [3]) which we wi ue i ad impify: U./.x/ 2 0=2 0=2 x 2 U./.x/ 0=2 0h=2 h. 1/. 2/ 2 2 h hc.. 1/. 2/ 2.. 1/ 0. U 2h.x/; h 1. 1/. 2/ 2 2 x 2. 2/ 2 2 U 2 2h.x/ h U 2.x/ 1 2 U 2.x/: (1) 1 We compute the um over eparatey:. 1/. 2/ /. /Š. /Š Š. /Š. /Š Š. 1 /Š. C 1/Š 0. /Š. 1/. C 1/Š C 1. /Š. C 1/Š /. / C 1 C. C 1/ C 2 C /. / C. C 1/ 1. / C. C 1/ C 1 1. / C 1. C 1/ C. C 1/. / C 1. 2 C 1/: I thi computatio oy the Vadermode covoutio formua [1] wa ued. Puggig thi formua ito (1) yied the aouced formua from the itroductio.
3 1158 H. Prodiger 3 Scaed Fiboacci umber: a imiar aayi I the very recet paper [2], the foowig poyomia have bee ivetigated: Oa.ı/ F 1. ı/ ; 0 Ob.ı/ F. ı/ : 1 1 For our purpoe, the -th poyomia houd have degree. Therefore we coider the foowig ight variatio: a.x/ F C1. x/ ; 0 C 1 b.x/ F. x/ 1 : Our goa, a before, i to expre the derivative of the poyomia by the poyomia themeve. Sice both famiie are a bai for the vector pace of poyomia, thi ca be achieved i a uique way. We wi wor out the correpodig coefficiet i the eque. Athough it i ot eeded, we give the doube geeratig fuctio: To chec thi i impe: t a.x/ 1 C.x 2/t C.1 x x 2 /t ; 2 t 1 b.x/ 1 C.x 2/t C.1 x x 2 /t : 2 t a.x/ t F C1. x/ t./ F C1. x/ C1 0 1 ˇ 1 z z 2 1 C.x 2/t C.1 x x 2 /t ; 2 1 ˇz tx a predicated, the other formua beig imiar. ow we eed to ivert: we ee the uique coefficiet uch that c ; a.x/ x ; 0 d ; b.x/ x : 0 They are give by c ; 1 F. 1/ ad d ; 1 F. 1/ C1. The proof that thi wor are traightforward: 0 1 F. 1/ 1 F 1 F The other oe i imiar: 1. 1/ C 1 C1 F C C 1 F. x/ 1. 1/. 1/ ŒŒ x :
4 Repreetig derivative of Chebyhev poyomia159 1 F. x/ 1 C 1. 1/ C 1 F C F. x/ 1 C 1. 1/ 1 ŒŒ C 1 x : F 1 ow we coider the -th derivative: a./.x/ F C1. 1/ Š. /Š x F C1. 1/ Š c ; a.x/. /Š 0 C. C /Š F CC1. 1/ c ; a.x/ C Š 0 0 C. C /Š 1 a.x/ F CC1. 1/. 1/ C Š F C1 0 Š a.x/. 1/ F CC1. 1/ C F C1 0 Š. 1/ a.x/ F CCC1. 1/ : F CC1 0 I order to impify the ier um, et u write W ad aume that 1, ice the itace 0 i differet (ad trivia). otice that. 1/ 0: The / F CCC1 F CC1 0 0 F 0 0 We eave it a a chaege to impify thi -th differece eve further. Ad ow we differetiate the other poyomia:. 1/ F CC1F C1 C F C F F CC1. 1/ F C F C1 C F F CC1. 1/ F C : F CC1 b./.x/ C 1 F. 1/ 1. 1/Š. 1 /Š x 1 C1 C 1 F. 1/ 1. 1/Š 1 d 1 ; b.x/. 1 /Š C1 0 C 1 C. C /Š F CC1. 1/ d ; b.x/ C C 1 Š 0 0 C 1 C. C /Š 1 b.x/ F CC1. 1/. 1/ C 1 C C 1 Š F C1 C 1 0
5 1160 H. Prodiger Š b.x/. 1/ C 1 C 1 0 Š. 1/ 0 b.x/. 1/ C 1 C 1 The ier um ca be reduced to the computatio of 0 C 1 C C / C C 1 C C C 1 but we have itte hope that thi ca be tured ito omethig ice. C C 1 C C C 1 F C F CC1 ; F CC1. 1/ C F C1 F CCC1 F CC1. 1/ : Referece [1] Graham R.L., Kuth.E., Patahi O., Cocrete Mathematic, 2d ed., Addio-Weey, Readig, [2] Hetmaio E., Piate B., Witua R., Biomia traformatio formuæ for caed Fiboacci umber, Ope Mathematic, 2017, 15, [3] Riorda J., Combiatoria Idetitie, Krieger, Hutigto, [4] Siyi W., Some ew idetitie of Chebyhev poyomia ad their appicatio, Advace i ifferece Equatio, 2015, 355 (8 page).
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