Properties of the second-kind Chebyshev polynomials of complex variable

Ìåòîäè ðîçâèíåííÿ ôóíêöié ó ðÿäè çà ñèñòåìàìè ôóíêöié äiéñíî ̈ ÷è êîìïëåêñíî ̈ çìiííî ̈ åôåêòèâíî âèêîðèñòîâóþòü ïðè ðîçâ'ÿçóâàííi áàãàòüîõ çàäà÷, ó òîìó ÷èñëi ïðèêëàäíîãî õàðàêòåðó. Çíà÷íîãî ðîçâèòêó íàáóëè ìåòîäè ðîçâ'ÿçàííÿ äèôåðåíöiàëüíèõ ðiâíÿíü, ùî ðóíòóþòüñÿ íà ðîçâèíåííÿõ ôóíêöié ó ñòåïåíåâi ðÿäè, à òàêîæ â ðÿäè çà ñèñòåìàìè îðòîãîíàëüíèõ ïîëiíîìiâ òà iíøèõ îðòîãîíàëüíèõ ôóíêöié îäíi1 ̈ ÷è äåêiëüêîõ çìiííèõ. Óçàãàëüíåííÿì ìåòîäó ðîçâèíåííÿ ôóíêöié ó ñòåïåíåâi ðÿäè 1 ̈õí1 ðîçâèíåííÿ çà ñèñòåìîþ ïîëiíîìiâ, áiîðòîãîíàëüíèõ ç äåÿêîþ iíøîþ ñèñòåìîþ ôóíêöié, ÿêi íàçèâàþòü àñîöiéîâàíèìè. Çà ïåâíèõ óìîâ äëÿ áóäü-ÿêî ̈ íåçàëåæíî ̈ i ïîâíî ̈ ñèñòåìè ôóíêöié ìîæíà ïîáóäóâàòè âiäïîâiäíó ñèñòåìó àñîöiéîâàíèõ ôóíêöié i êîíñòðóþâàòè ðÿäè çà íåþ. Ó äàíié ðîáîòi äîñëiäæåíî âëàñòèâîñòi ïîëiíîìiâ ×åáèøîâà äðóãîãî ðîäó êîìïëåêñíî ̈ çìiííî ̈. Îòðèìàíî iíòåãðàëüíi çîáðàæåííÿ öèõ ïîëiíîìiâ. Ïîáóäîâàíî àñîöiéîâàíi ôóíêöi ̈, áiîðòîãîíàëüíi íà çàìêíåíèõ êðèâèõ êîìïëåêñíî ̈ ïëîùèíè ç ïîëiíîìàìè ×åáèøîâà äðóãîãî ðîäó. Âñòàíîâëåíî äîñòàòíi óìîâè, çà ÿêèõ àíàëiòè÷íi ôóíêöi ̈ ìîæíà ðîçêëàñòè â ðÿäè çà öi1þ ñèñòåìîþ ïîëiíîìiâ. Íàâåäåíî ïðèêëàäè ðîçêëàäiâ ôóíêöié â ðÿäè çà ðîçãëÿäóâàíîþ ñèñòåìîþ ïîëiíîìiâ ó êîìïëåêñíèõ îáëàñòÿõ. Êðiì òîãî, îòðèìàíî êîìáiíàòîðíi òîòîæíîñòi, ÿêi ìîæíà âèêîðèñòîâóâàòè äëÿ iíøèõ äîñëiäæåíü. Êëþ÷îâi ñëîâà: ìíîãî÷ëåíè ×åáèøîâà, àíàëiòè÷íi ôóíêöi ̈, áiîðòîãîíàëüíi ñèñòåìè ôóíêöié, àñîöiéîâàíi ôóíêöi ̈. We construct a system of functions biorthogonal with Chebyshev polynomials of the second kind on closed contours in the complex plane. Properties of these functions and su cient conditions of expansion of analytic functions into series in Chebyshev polynomials of the second kind in complex domains are investigated. The examples of such expansions are given. In addition, combinatorial identities of self-interest are obtained.


Introduction
he gheyshev polynomils form the sis of theoretil nd prtil studies of funtion pproximtion theoryF hey re e'etively used in the prolems of omputtiE onl mthemtisD for solving di'erentil nd integrl equtionsD for numeril diE 'erentition nd integrtionF roperties of the gheyshev polynomils of rel vrile re su0iently well investiE gted in ID PF gonsiderly fewer reserhes onern properties of these polynomils in omplex dominsF ome properties of gheyshev polynomils in omplex plneD s well s expnsions of nlyti funtions y the gheyshev polynomils of the (rst kind in omplex domins re otined in PF qenerliztion of the method of expnsions of funtions into power series is their expnsions in system of polynomilsD iorthogonl with some other system of funtiE ons whih re lled ssoited funtionsF nder ertin onditionsD for ny linerly independent nd omplete system of funtionsD it is possile to onstrut orrespondiE ng system of ssoited funtions nd onstrut series fter itF sn this pperD we onstrut system of funtions iorthogonl with gheyshev polynomils of the seond kind on losed ontours in omplex plneD nd estliE sh onditions under whih nlyti funtions n e expnded into series in these polynomilsF ixmples of suh expnsions re givenF sn dditionD omintoril identiE ties of selfEinterest re otinedF vet9s denote y U n (z) the gheyshev polynomils of the seond kind of omplex vrileF por themD expliit formuls QD pFIVT U n (z) = n 2 k=0 (−1) k 2 n−2k C k n−k z n−2k , n = 0, 1, . . . , @IFIA holdF rere nd elowD C m n e the inomil oe0ientsF sing reltions @IFIA one n otin the expressions of the polynomils U n (z) for odd nd even vlues of nD respetivelyX U 2n (z) = n k=0 (−1) k 2 2(n−k) C k 2n−k z 2(n−k) = n l=0 (−1) n−l 2 2l C 2l n+l z 2l , @IFPA Theorem 1. The polynomials U n (z) can be represented as follows: Proof. sing the xewton inomil formul nd hnging the order of summtionD we (nd from @PFIA king into ount the omintoril identity we otin reltion @IFIAF vet Γ R denote the ellipse given y the eqution nd D R e domin whose oundry is the ellipse Γ R F Theorem 2. The polynomials U n (z) satisfy the following estimates: Proof. sing the inequlity x 2 − x 2 t 2 + t 2 ≤ 1D whih holds for ll x nd t suh tht |x| ≤ 1D |t| ≤ 1D from reltion @PFIA it follows tht he polynomil 1 n + 1 U n (z) stis(es the onditions of heorem in RD pFITTX if for the polynomial W n (z) of degree n on the real interval [−1; 1] the inequality |W n (z)| ≤ M holds, where M = const, then for any z outside this interval the following estimate is valid where a and b are axes of an ellipse passing through the point z with its focuses at points z = ±1. reneD sine the xes of the ellipse Γ r D 1 < r ≤ RD with eqution @PFQA re 1 2 r + 1 r nd 1 2 r − 1 r D respetivelyD then estimte @PFSA immeditely followsF Theorem 3. For the Chebyshev polynomials, integral representation Proof. epplying the xewton inomil formul to z + t √ z 2 − 1 n nd using the expnsion ϕ n (t) = ∞ j=0 n + 1 2j + 1 1 t 2j+1 D ording to @PFTAD we otin es usulD δ mn = 0, m = n, 1, m = n is the uroneker deltF st is known tht SD ppFVIEVP where L is n ritrry losed ontour enveloping the point a nd one running in positive diretionF herefore U n (z) = (n + 1) sing the xewton inomil formul nd hnging the order of summtionD we hve U n (z) = (n + 1) reneD tking into ount omintoril identity @PFPAD we otin @IFIAF st is known PD pFPV tht the monomil of z n n e uniquely expressed in terms of gheyshev polynomils U n (z)D s followsX prom reltions @PFVA we n otin expressions of z n in the se of odd nd even vlues of nD respetivelyX . @PFIHA Theorem 4. The combinatorial identities hold: Proof. ustituting expressions @PFWA into @IFPA nd hnging the order of summtionD we (nd tht reneD using the independene of the polynomils U n (z)D we otin identity @PFIIAF fy nlogyD sustituting expressions @PFIHA into @IFQAD we otin identity @PFIPAF vet A R denote the spe of holomorphi funtions in the disk |z|<RD where 0 < R ≤ ∞F Theorem 5. The system of polynomials {U n (z)} ∞ n=0 is linearly independent and complete in the space A R .
Proof. ine the oe0ient of the term of the highest degree of the polynomil U n (z) is nonzeroD then the system {U n (z)} ∞ n=0 is linerly independent TD pFIQUF sn dditionD it is omplete TD pFIQUD euse every power z n n e uniquely expressed s liner omintion of the polynomils U n (z)F

Functions associated with polynomials
ghnging the order of summtionD we see tht fy nlogy with TD pFIPH de(ne the funtions ω m (z) ssoited with the polynomiE ls U n (z)X sing reltion @QFRAD expressions of the ssoited funtions ω m (z) for even nd odd vlues of the indexes m n e written sX where C is positively oriented circle |z| = q, 1 < q < R.
Proof. prom the symptotil formul n! ∼ n n e n D n → ∞D it follows tht where L is losed positively oriented ontour ontining the singulr points of funtiE ons ω k (z)F Theorem 7. The system of associated functions {ω n (z)} ∞ n=0 is biorthogonal to the system of polynomials U n (z) along any piecewise-smooth closed contour γ enveloping the disk |z| ≤ 1, i. e.
st is known PD pFIRW tht where T k (x) is the kEth degree gheyshev polynomil of the (rst kindY . . + a n T n (x) + . . .
Proof. ustituting expressions @QFSA nd @QFTA into the rightEhnd side of expnsion ghnging the order of summtion in the lst two sums nd tking into ount @PFWA nd @PFIHAD we (nd tht xowD let us show tht the series in @QFIUA is uniformly onvergent for t ∈D ∞ ρ D z ∈D 0 r F rere ρ nd r re numers suh tht 0 < r < ∞D ρ > max{1, r}F yn the sis of @PFSA nd @QFIQAD we hve ine ρ > rD the ove series onvergesF hereforeD the series in @QFIUA onverges uniE formly in the ove mentioned dominsF 4. Expansion of functions into series in polynomials U n (z) Theorem 11. Let f (z) be a function of complex variable which is holomorphic in an open domain D R whose boundary is ellipse Γ R , 1 < R ≤ ∞, with equation @PFQA and bounded by M in Γ R , i. e.
Then the series ∞ n=0 L n (f )U n (z) @RFPA converges uniformly in the closed domain D r whose boundary is the ellipse Γ r , where 1 ≤ r < R.