2 JOAQUIN BUSTO AND SERGEI K. SUSLOV l (.8) l cos nx l sin mx l dx m 6 n: In th prsnt papr w discuss a -vrsion of th Fourir sris (.) with th aid of ba

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1 BASIC ANALOG OF FOURIER SERIES ON A -QUADRATIC GRID JOAQUIN BUSTO AND SERGEI K. SUSLOV Abstract. W prov orthogonality rlations for som analogs of trigonomtric functions on a -uadratic grid introduc th corrsponding -Fourir sris. W also discuss svral othr proprtis of this basic trigonomtric systm th -Fourir sris. A priodic function with priod 2l, (.) can b rprsntd as th Fourir sris, (.2) whr (.3) (.4) (.5) f(x) a +. Introduction f(x + 2l) f(x) X n a 2l a n l b n l a n cos n l x + b n sin n l x l l l l l l f(x) dx f(x) cos n l x dx f(x) sin n l x dx: For convrgnc conditions of (.2) s, for xampl, [], [28], [3]. Th formulas (.3){(.5) for th cocints of th Fourir sris ar consuncs of th orthogonality rlations for trigonomtric functions (.6) (.7) l l l l cos nx l sin nx l cos mx l sin mx l dx m 6 n dx m 6 n Dat: May 2, Mathmatics Subjct Classication. Primary 33B, 33D5 Scondary 42C. Ky words phrass. Trigonomtric functions, basic trigonomtric functions, orthogonality rlations, Fourir sris, -Fourir sris.

2 2 JOAQUIN BUSTO AND SERGEI K. SUSLOV l (.8) l cos nx l sin mx l dx m 6 n: In th prsnt papr w discuss a -vrsion of th Fourir sris (.) with th aid of basic or -analogs of trigonomtric functions introducd rcntly in [5] (s also [5] [24]). Our rst main objctiv will b to stablish analogs of th orthogonality rlations (.5){(.7) for -trigonomtric functions on a -uadratic grid. Thr ar svral ways to prov th orthogonality rlations (.6){(.8) for trigonomtric functions. Th mthod basd on th scond ordr dirntial uation, (.9) u +! 2 u can b xtndd to th cas of basic trigonomtric functions. Considr, for xampl, two functions cos!x cos! x, which satisfy (.9) with dirnt ignvalus!!. Thn, (.) (! 2! 2 ) l l cos!x cos! x dx Th right sid of (.) vanishs whn W (cos!x cos! x) j l l cos!x cos! x! sin!x! sin! x l l : (.) sin!l sin! l which givs (.2)! l n! l m whr n m 2 3 :::. In th sam mannr, on can prov (.7). Th last uation (.8) is valid by symmtry. W shall xtnd this considration to th cas of th basic trigonomtric functions in th prsnt papr. This papr is organizd as follows. In Sction 2 w introduc th -trigonomtric functions. In th nxt sction w driv a continuous orthogonality proprty of ths functions, thn, in Sction 4, w formally discuss th limit! of ths nw orthogonality rlations. Sction 5 is dvotd to th invstigation of som proprtis of zros of th basic trigonomtric functions in Sction 6 w valuat th normalization constants in th orthogonality rlations for ths functions. In Sction 7 w stat th orthogonality rlation for th corrsponding -xponntial functions. Finally, w introduc basic analogs of Fourir sris in Sction 8, in Sctions 9{ w giv a proof of th compltnss of th -trigonomtric systm stablish som lmntary facts about convrgnc of our -Fourir sris. Exampls of ths sris ar considrd in Sctions 2 4 w prov som basic trigonomtric idntitis w hd in Sction 3. Som miscllanous rsults concrning -trigonomtric functions ar discussd in Sction 5.

3 -FOURIER SERIES 3 2. Analogs of Trigonomtric Functions on a -Quadratic Grid Th following functions C(x) S(x) givn by (2.) (2.2) C(x) C (x!) (!2 2 ) (! 2 2 ) 2 ' S(x) S (x!) (!2 2 ) 2 4! (! 2 2 ) 2 ' 2 3 cos 2 2! 2 2! 2 wr discussd rcntly [5], [5] [24] as -analogs of cos!x sin!x on a - uadratic lattic x cos. Ths functions ar spcial cass y of mor gnral basic trigonomtric functions (2.3) (2.4) C(x y) C (x y!) (!2 2 ) (! 2 2 ) 4 ' 3 2 i+i' 2 ii' 2 i'i 2 ii' 2 2! 2 S(x y) S (x y!) (!2 2 ) 2 4! (! 2 2 ) (cos + cos ') 4 ' 3 i+i' ii' i'i ii' 32 32! 2 which ar -analogs of cos!(x + y) sin!(x + y), rspctivly (s [24]). Hr x cos y cos '. Usually w shall drop from th symbols C (x!), S (x!), C (x y!), S (x y!) bcaus th sam bas is usd throughout th papr. Th symbols 2 ' 4 ' 3 in (2.){(2.4) ar, of cours, spcial cass of basic hyprgomtric functions, (2.5) r' s a a 2 ::: a r b b 2 ::: b s t

4 4 JOAQUIN BUSTO AND SERGEI K. SUSLOV : X n (a a 2 ::: a r ) n ( b b 2 ::: b s ) n () n n(n)2 +sr t n : Th stard notations for th -shiftd factorial ar (2.6) (2.7) (a ) : (a ) n : (a a 2 ::: a m ) n : n Y a k k my l (a l ) n whr n 2 ::: or, whn jj <. S [7] for an xcllnt account of th thory of basic hyprgomtric functions. Functions (2.){(2.4) ar dnd hr for j!j < only. For an analytic continuation of ths functions in a largr domain s [2], [5], [24]. For xampl, (2.8) (2.9) C(x) S(x)!2! 2 2 (! 2 2 ) 2 ' 2!2! 2! 2! 2 2 2! 2 2! ! ( 3! 2 2 ) cos 2 ' 2!2! 2 2! 2 2! : On can s from (2.8) (2.9) that th basic trigonomtric functions (2.) (2.2) ar ntir functions in z whn i z. Analytic continuation of -trigonomtric functions (2.3) (2.4) can b obtaind on th basis of th \addition" thorms, C (x y) C (x) C (y) S (x) S (y) S (x y) S (x) C (y) + C (x) S (y) found in [24]. Th basic trigonomtric functions (2.){(2.4) ar solutions of a dirnc analog of uation (.9) on a -uadratic lattic, (2.) rx (z) ru(z) rx(z) + u(z) whr x(z) 2 (z + z ), z i, x (z) x(z + 2), 4 2! 2 ( ) 2, f(z) rf(z + ) f(z + ) f(z). S [5], [5], [2], [23], [24] for mor dtails. Euation (2.) can also b rwrittn in a mor symmtric form, (2.) x(z) u(z) x(z) whr f(z) f(z + 2) f(z 2) ( ) 2!2 u(z)

5 -FOURIER SERIES 5 Th -trigonomtric functions (2.){(2.4) satisfy th dirnc-dirntiation formulas (2.2) (2.3) x x C(x y) 24 S(x y) 24! S(x y)! C(x y): S [5] [24]. Applying th oprator x to th both sids of (2.2) or (2.3) w obtain uation (2.) again. Euation (2.) is a vry spcial cas of a gnral dirnc uation of hyprgomtric typ on nonuniform lattics (cf. [5], [2], [23], [24]). Th Asky{Wilson polynomials thir spcial limiting cass [4], [7], [2] ar wll-known as th simplst th most important orthogonal solutions of this dirnc uation of hyprgomtric typ. Rcntly, Ismail, Masson, Suslov [2], [3], [25], [26] hav found anothr typ of orthogonal solutions of this dirnc uation. In th prsnt papr w shall discuss this nw orthogonality proprty at th lvl of basic trigonomtric functions. 3. Continuous Orthogonality Proprty for -Trigonomtric Functions Our main objctiv in this papr is to nd th orthogonality rlations for - trigonomtric functions (2.){(2.2) similar to th orthogonality rlations (.5){(.7). Considr dirnc uations for th functions u(z) C (x(z)!) v(z) C (x(z)! ) in slf-adjoint form, (3.) (3.2) rx (z) (z) ru(z) rx(z) (z) rv(z) rx (z) rx(z) + (z)u(z) + (z)v(z) whr th function (z) satiss th \Parson uation"[2], [23], (3.3) (3.4) (z + ) (z) On can asily chck that (3.5) (z + ) (z) (z) (z + ) 4z2 2 2z+ 42 ( ) 2!2 42 ( ) 2!2 : 4z2 for (z) (2z 2z ) z z

6 6 JOAQUIN BUSTO AND SERGEI K. SUSLOV (3.6) (z + ) (z) 4z+2 for (z) 2+2z 22z 22+2z 222z 2 (cf. [2], [25], [26]). Thrfor, w can choos th following solution of (3.3), (3.7) (z) ( 2z 2z ) ( z z ) ( 2+2z 22z 22+2z 222z 2 ) whr is an arbitrary additional paramtr. W shall s latr that this solution satiss th corrct boundary conditions for our scond ordr dividd-dirnc Asky{Wilson oprator (2.) for crtain valus of this paramtr. Lt us multiply (3.) by v(z), (3.2) by u(z) subtract th scond uality from th rst on. As a rsult w gt (3.8) whr (3.9) ( ) u(z) v(z) (z)rx (z) [(z) W (u(z) v(z))] W (u(z) v(z)) u(z) ru(z) rx(z) v(z) rv(z) rx(z) u(z) rv(z) ru(z) v(z) rx(z) rx(z) is th analog of th Wronskian [2]. Intgrating (3.8) ovr th contour C indicatd in th Figur on th nxt pag whr z is such that z i log 2 32 givs (3.) ( ) C C u(z) v(z) (z)rx (z) dz [(z) W (u(z) v(z))] dz: As a function in z, th intgr in th right sid of (3.) has th natural purly imaginary priod T log whn < <, so this intgral is ual to (3.) D (z) W (u(z) v(z)) dz whr D is th boundary of th rctangl on th Figur orintd countrclockwis. Th basic trigonomtric functions C(x) S(x) ar ntir functions in th complx z-plan du to (2.8){(2.9). Thrfor, th pols of th intgr in (3.) insid th rctangl in th Figur ar th simpl pols of (z) at z, z at z i log, z i log whn < R < 2. Hnc, by Cauchy's thorm, (3.2) W (u v) dz D Rs f(z)j z + Rs f(z)j z + Rs f(z)j zi log + Rs f(z)j zi log

7 -FOURIER SERIES 7 Im z 3i 2 log 6 3i 2 log + r r C + i log + i log r r R z i 2 log i 2 log + whr (3.3) f(z) (z) W (u(z) v(z)) z ( 2z 2z ) W (u(z) v(z)) : ( 2+2z 22z 22+2z 222z 2 ) Evaluation of th rsidus at ths simpl pols givs (3.4) Rs f(z)j z lim z! (z ) f(z) ( 2 2 ) 2 log ( ) W (u(z) v(z))j z (3.5) Rs f(z)j z lim (z + ) f(z) z! ( 2 2 ) 2 log ( ) W (u(z) v(z))j z

8 8 JOAQUIN BUSTO AND SERGEI K. SUSLOV (3.6) (3.7) Rs f(z)j zi log lim (z + i log ) f(z) z!i log ( 2 2 ) 2 log ( ) W (u(z) v(z))j zi log Rs f(z)j zi log lim (z + + i log ) f(z) z! But (3.8) by (3.9), thrfor, (3.9) W (u(z) v(z)) ( 2 2 ) 2 log ( ) W (u(z) v(z))j zi log : v(z)u(z ) u(z)v(z ) x(z) x(z ) W (u(z) v(z))j z W (u(z) v(z))j z W (u(z) v(z))j zi log W (u(z) v(z))j zi log du to th symmtris C(x) C(x), x(z) x(z), x(z) x(z i log ). Thus, th rsidus ar ual as a rsult w gt (3.2) 2 ( ) 2!2! 2 C u(z) v(z) (z)rx (z) dz i ( 2 2 ) log ( ) W (u() v()) whr < R < 2. W hav stablishd our main uation (3.2) for th cas u(z) C (x(z)!) v(z) C (x(z)! ). Th sam lin of considration shows that this uation is also tru whn u(z) S (x(z)!) v(z) S (x(z)! ). Th corrsponding analogs of th Wronskians in (3.2) can b writtn as (3.2) W (C(x(z)!) C(x(z)! )) (3.22) 24 [! C (x(z)! ) S (x(z 2)!)! C (x(z)!) S (x(z 2)! )] W (S(x(z)!) S(x(z)! )) 24 [! S (x(z)!) C (x(z 2)! )! S (x(z)! ) C (x(z 2)!)]

9 -FOURIER SERIES 9 by (2.2){(2.3), rspctivly. On can s from (3.2) (3.22) that th right sid of (3.2) vanishs in both cass whn ignvalus!! ar roots of th following uation (3.23) S (x(4)!) S (x(4)! ) : This is a dirct analog of (.) for basic trigonomtric functions. In th last cas, u(z) C (x(z)!) v(z) S (x(z)! ), th lft sid of (3.2) vanishs by symmtry. It is intrsting to vrify that by using our mthod as wll. Euations (3.) to (3.8) ar valid again. But now (3.24) W (u(z) v(z))j z W (u(z) v(z))j z W (u(z) v(z))j zi log W (u(z) v(z))j zi log du to th symmtris C(x) C(x), S(x) S(x), x(z) x(z), x(z) x(z i log ). Thrfor, (3.25) 2 ( ) 2!2! 2 C u(z) v(z) (z)rx (z) dz i ( 2 2 ) 2 log ( ) [W (u() v()) W (u() v())] whn < R < 2. Combining all th abov cass togthr, w nally arriv at th continuous orthogonality rlations for basic trigonomtric functions, (3.26) C (cos!) C (cos! ) 8 >< >: ( 2 2 ) d if! 6! 2 2 ( ) 2 S (!) if!! (3.27) (3.28) S (cos!) S (cos! ) 8 >< >: ( 2 2 ) d if! 6! 2 2 ( ) 2 C (cos!) S (cos! ) S (!) if!! ( 2 2 ) d : Hr : x(4) th ignvalus!! satisfy th \boundary" condition (3.23).

10 JOAQUIN BUSTO AND SERGEI K. SUSLOV For arbitrary! 6! on gts from (3.2){(3.22) (3.29) (3.3) C (cos!) C (cos! ) ( 2 2 ) d 2 ( ) 2 [! C (! ) S (!)! C (!) S (! )] Also, in th limit!!!, (3.3) (3.32)! 2! S (cos!) S (cos! ) ( 2 2 ) d 2 ( ) 2 [! S (!) C (! )! S (! ) C (!)] :! 2! C 2 (cos!) 2 2! ( ) 2 d ( 2 2 )! C S + C (!) C (!) S (!) S 2 (cos!) 2 2! ( ) 2 d ( 2 2 )! C S C (!) C (!) S (!) : W rmind th radr that is dnd by x(4) notation will b usd throughout this work. This 4. Formal Limit! In this sction w formally obtain orthogonality of th trigonomtric functions as limiting cass of our orthogonality rlations (3.26){(3.28) for basic trigonomtric functions. According to [24], (4.) lim! C x y 2! ( ) cos! (x + y)

11 (4.2) lim S x y! If! 6! w can rwrit (3.26) as (4.3) whr -FOURIER SERIES 2! ( ) sin! (x + y) : C (cos!) C (cos! ) 2 d (4.4) Using th limiting rlation [7] (a r) : (a r) (ar r) : (4.5) on can s that (4.6) lim (a r) (! a) 2! 2 sin as!. Thrfor, changing! to ( )!2 in (4.3), with th hlp of (4.) whn y w obtain th orthogonality rlation (.6) with l. Th boundary condition (.) follows from (3.23) in th sam limit. Whn!! w can rwrit (3.26) as (4.7) C 2 (cos!) 2 2 ( ) 2 (2) C (!) S whr (4.8) (z) ( ) z ( ) ( z ) is a -analog of Eulr's gamma function (z) (s, for xampl, [7]). Changing! to ( )!2 in (4.8), with th aid of (4.9) w gt (4.) 2 cos 2 nx dx lim (z) (z)! 2 2 (2) cos2 n 2 whr n 2 :::, in th limit!. In a similar mannr on can obtain (.7) (.8) from (3.27) (3.28), rspctivly.

12 2 JOAQUIN BUSTO AND SERGEI K. SUSLOV 5. Som Proprtis of ros In Sction 3 w hav stablishd th orthogonality rlations for th basic trigonomtric functions (3.26){(3.28) undr th boundary condition (3.23). Hr w would lik to discuss som proprtis of!-zros of th corrsponding basic sin function, (5.) S (!) (!2 2 ) (! 2 2 )! 2 th basic cosin function, 2 ' ! 2 3 ( 3! 2 2 )! 2 2! 2 2 ' 2!2! 2 32! 2 52! (5.2) C (!) (!2 2 ) (! 2 2 ) 2 2 ' 32 2! 2 2! 2 (! 2 2 ) 2 ' 2!2! 2 2! 2 32! 2 2 On can s that ths functions hav almost th sam structur as th -Bssl function discussd in [2], [3]. So w can apply a similar mthod to stablish main proprtis of zros of th functions (5.){(5.2). Th rst proprty is that th -sin function S (!) has an innity of ral!-zros. To prov that w can again considr th larg!-asymptotics of th function (5.). Th 2 ' hr can b transformd by (III.) of [7], which givs : (5.3) S (!) 52 32! 2 2 ( 3! 2 2 )! 2 2 ' 2! 2 32! : For larg valus of!, such that! n whr n 2 :::, 2! (5.4) 2' ! 2! ' ( 2 )

13 by th -binomial thorm. Thrfor, as!!, -FOURIER SERIES 3 2 (5.5) S (!) ( 2 ) 2! 32! 2 2 (! 2 2 ) [ + o()] by (5.3) (5.4). But th function 32! 2 2 oscillats has an innity of ral zros as! approachs innity. Indd, considr th points! n, such that (5.6) 2 2 n 2 2n whr n 2 ::: 2 < 2 < 32, as tst points. Thn, by using (I.9) of [7], (5.7) S ( n ) ( 2 ) 2 ( 2 2 ) () n n n ( 2 2 ) n [ + o()] as n!, on can s that th right sid of (5.7) changs sign innitly many tims at th tst points! n as! approachs innity. In a similar mannr, on can prov that th -cosin function C (!) has an innity of ral!-zros also. Thus w hav stablishd th following thorm. Thorm 5.. Th basic sin S (!) basic cosin C (!) functions hav an innity of ral!-zros whn < <. Now w can prov our nxt rsult. Thorm 5.2. Th basic sin S (!) basic cosin C (!) functions hav only ral!-zros whn < <. Proof. Suppos that! is a zro of th basic sin function (5.) which is not ral. It follows from (5.) (III.4) of [7] that (5.8) S (!) 52! 2 2 (! 2 2 )! 2 2 ' ! ! 2 : Now w can s that! is not purly imaginary, bcaus othrwis our function would b a multipl of a positiv function. Lt! b th complx numbr conjugat to!, so that! is also a zro of (5.) bcaus this function is a ral function of!. Sinc! 2 6! 2 th intgral in th orthogonality rlation (3.26) uals zro, but th intgr on th lft is positiv,

14 4 JOAQUIN BUSTO AND SERGEI K. SUSLOV so w hav obtaind a contradiction. Hnc a complx zro! cannot xist. On can considr th cas of th basic cosin function in a similar fashion. Thorm 5.3. If < <, thn th ral!-zros of th basic sin S (!) basic cosin C (!) functions ar simpl. Proof. This follows dirctly from th rlations (3.3) (3.32). Considr, for xampl, th cas of th basic sin function. If!!, thn th intgral in th lft sid (3.3) is positiv, which mans that S (!) 6 whn S (!). Th sam tru for th zros of th basic cosin function. Our nxt proprty is that th positiv zros of th basic sin function S (!) ar intrlacd with thos of th basic cosin function C (!). Thorm 5.4. If!! 2! 3 ::: ar th positiv zros of S (!) arrangd in ascnding ordr of magnitud, $ $ 2 $ 3 ::: ar thos of C (!), thn (5.9) if < <.! < $ <! < $ 2 <! 3 < $ 3 < ::: Proof. Suppos that! k! k+ ar two succssiv zros of S (!). Thn S (!) has dirnt signs at!! k!! k+. This mans, in viw of (3.32), that C (!) changs its sign btwn! k! k+, thrfor, has at last on zro on ach intrval (! k! k+ ). To complt th proof of th thorm, w should show that C (!) changs its sign on ach intrval (! k! k+ ) only onc. Suppos that C ( $ k ) C ( $ k+ )! k < $ k < $ k+ <! k+. Thn, by (3.32), th function S (!) has dirnt signs at! $ k! $ k+, thrfor, this function has at last on mor zro on (! k! k+ ). So, w hav obtaind a contradiction,, thrfor, th basic cosin function C (!) has xactly on zro btwn any two succssiv zros of th basic sin function S (!). Th proof of Thorm 5. has strongly indicatd that asymptotically th larg!-zros of th basic sin function S (!) ar (5.)! n { n n 4 { n < 34 as n!. Th sam considration as in [] [3] shows that S (!) changs sign only onc btwn any two succssiv tst points! n! n+ dtrmind by (5.6) for larg valus of n. W includ dtails of this proof in Sction 6 to mak this work as slf-containd as possibl. Our nxt thorm provids a mor accurat stimat for th distribution of th larg zros of this function. Thorm 5.5. If!! 2! 3 ::: ar th positiv zros of S (!) arrangd in ascnding ordr of magnitud, thn (5.) as n!.! n 4n + o ()

15 Proof. In viw of (5.) (III.32) of [7], (5.2) S (!) -FOURIER SERIES 5! ! 2 2! 2 2 ( 3! 2 2! 2 2 ) + 2 ' ! 2! ! 2 2! 2 2 ( 3! 2 2! 2 2 ) 2 ' ! 2 which givs th larg!-asymptotic of S (!). Whn! 4n n 2 3 ::: th rst trm in (5.2) vanishs w gt (5.3) S 4n () n n n 2 ( 2 2 ) n with th hlp of (I.9) of [7], Thus, (5.4) which provs our thorm. 2 ' lim n! S 4n In a similar fashion, on can stablish th following thorm. 2 2n+2 Thorm 5.6. If $ $ 2 $ 3 ::: ar th positiv zros of C (!) arrangd in ascnding ordr of magnitud, thn (5.5) as n!. $ n 34n + o () Th asymptotic formulas (5.) (5.5) for larg!-zros of th basic sin S (!) basic cosin C (!) functions conrm th intrlacing proprty (5.9) from Thorm 5.4. Lt us also discuss th larg!-asymptotics of th basic sin S (x!) basic cosin C (x!) functions whn x cos blongs to th intrval of orthogonality < x <. From (2.) (2.2) on gts (5.6) C (cos!) 2 ' 2! 2! 2! 2 2 ( 4i! 2! 2 2 ) 2 ' 2 4i 2 2! 2

16 6 JOAQUIN BUSTO AND SERGEI K. SUSLOV +! 2! 2 2 ( 4i! 2! 2 2 ) 2 ' 2 4i 2 2! 2 (5.7) S (cos!) 24! cos 2 ' 24! cos 3 2! 2 " 2 2! 2! 2 2 ( 3 4i! 2! 2 2 ) 2 ' 2 4i 2 2! ! 2! 2 2 ( 3 4i! 2! 2 2 ) # 2 ' 2 4i 2 2! 2 (5.8) C (cos!) by (III.3) (III.32) of [7]. For jxj <, jj < larg! it is clar from (5.6) (5.7) that th lading trms in th asymptotic xpansions of C (cos!) S (cos!) ar givn by!2 2 ( 4i 2 ) (! 2 2 ) + ( 4i 2 )!2 2 (! 2 2 ) (5.9) S (cos!) 24! cos " 2! 2 2 ( 3 4i 2 ) (! 2 2 ) 2! 2 2 # + ( 3 4i 2 ) (! 2 2 )

17 -FOURIER SERIES 7 rspctivly. In particular, whn!! n ar larg zros of th basic sin function S (!) w can stimat (5.2) (5.2) C (cos! n ) C cos 4n S (cos! n ) S cos 4n du to (5.) as n!. Rlations (5.8){(5.2) lad to th following thorm. Thorm 5.7. For < x cos < jj < th lading trm in th asymptotic xpansion of C (cos! n ) as n! is givn by (5.22) whr (5.23) (5.24) (5.25) C cos 4n 2 2 ( 2 ) 2 A i cos ((2 + ) n ) A i ( ) A i 2 ( 2 2 ) arg A i : For < x cos < jj < th lading trm in th asymptotic xpansion of S (cos! n ) as n! is givn by (5.26) whr S cos 4n 2 2 ( 2 ) 2 B i cos ((2 + ) (n ) ) (5.27) (5.28) (5.29) From (5.23) (5.27), B i i ( ) B i 2 ( 2 2 ) arg B i : (5.3) A i i ( ) B i :

18 8 JOAQUIN BUSTO AND SERGEI K. SUSLOV It is worth mntioning also that th factor A i 2 B i 2 coinsids with th wight function in our orthogonality rlations (3.26){(3.28) for th basic trigonomtric functions. In a similar fashion, on can us th rst lins in (5.6), (5.7), Exrcis 3.8 of [7] (s also th sam lin of rasonings in [8]) to stablish complt asymptotic xpansions of th basic sin cosin functions for th larg valus of!. Thorm 5.8. For < x cos < jj < complt asymptotic xpansions of C (cos!) S (cos!) as j!j! ar givn by!2 2 (5.3) C (cos!) ( ) (! 2 2 ) X 2n 2n!2 2 ( 2 2 4i 2 ) n n!2 2 n + ( ) (! 2 2 ) X n 2n 2n!2 2 ( 2 2 4i 2 ) n n (5.32) S (cos!) i 2! 2 2 ( ) (! 2 2 ) X 2n+4 2n 2! 2 2 ( 2 2 4i 2 ) n n n + i 2! 2 2 ( ) (! 2 2 ) X 2n+4 2n!2 2 ( 2 2 4i 2 ) : n n n Th asymptotic xpansions (5.3){(5.32) ar not in trms of th usual asymptotic sunc (x!) n, but ar sums of two complt asymptotic xpansions in trms n (cf. [8]). of th \invrs gnralizd powrs" 2! 2 2 Rmark 5.. Mourad Ismail pointd out to our attntion th following uadratic transformation formula (5.33) 2' r 2 4 ( ) ( + ) (2r) (r ) J (2) (r ) whr jrj < 2, rlating th 2 ' of a givn structur with Jackson's basic Bssl functions J (2) (r ). A similar rlation was arlir found by Rahman [22]. This n

19 -FOURIER SERIES 9 transformation shows that our basic sin S (!) basic cosin C (!) functions ar just multipls of J (2) (2) (2! ) J (2! ), namly, 2 2 (5.34) (5.35) S (!) ( )! 2 J (2) (2! ) ( 2 ) (! ) C (!) ( )! 2 J (2) (2! ) : ( 2 ) (! ) Th main proprtis of zros of th -Bssl functions J (2) (r ) wr stablishd in Ismail's paprs [9] [] by a dirnt mthod. This givs indpndnt proofs of our Thorms 5.{5.4. Som monotonicity proprtis of zros of J (2) (r ) wr discussd in [4]. Chn, Ismail, Muttalib [8] hav found a complt asymptotic xpansion of J (2) (r ) for th larg argumnt, (5.36) J (2) (r ) 2 2 ( ) r 2 " i r 2 (+2)2 2 X +2 n2 n ( ) n n + i r 2 (+2)2 2 X n n2 +2 n ( ) n i r 2 (+2)2 2 i r 2 (+2)2 2 This follows also from Exrsiss of [7]. Euations (5.34){(5.36) rsult in (5.) (5.5). 6. Evaluation of Som Constants In this sction w shall nd xplicitly th valus of th normalization constants in th right sids of th orthogonality rlations (3.26){(3.27) for th basic sin basic cosin functions. First, w valuat th intgral (6.) 2k (!) C 2 (cos!) + S 2 (cos!) d ( 2 2 ) C (cos cos!) whr w hav usd th idntity (4.4) of [24], (6.2) ( 2 2 ) d C (x x!) C 2 (x!) + S 2 (x!) : n n # :

20 2 JOAQUIN BUSTO AND SERGEI K. SUSLOV In viw of (2.3), for j!j < on can writ (6.3) 2 (!2 2 ) (! 2 2 ) k (!) X! 2 n n 2 n ( 2 ) n ( n+2 n+2 ) d: Th intgral in th right sid is a spcial cas of th Asky{Wilson intgral [4], (6.4) Thrfor, ( n+2 n+2 ) d 2 ( 2n+2 ) ( n+2 n+ n+ n+ n+ n+32 ) : (6.5) 2 (!2 2 ) (! 2 2 ) k (!) 2 2 ( 2 ) X (! 2 ) n n+2 whr w hav usd th idntity 2n+2 n+ n+ n+32 n+32 : But, X (! 2 ) n n+2 n n 2 2' 2! ! 2 ( 2! 2 ) 2' 2! 2 2! 2 by (III.) of [7]. Th last lin provids an analytic continuation of this sum in th complx!-plan. Finally, w obtain (6.6) k (!) 2 C 2 (cos!) + S 2 (cos!) d ( 2 2 ) (6.7) 2 2! 2 (! 2 2 ) (! 2 ) (! 2 2 ) 2 ' 2! 2 2! 2 :

21 -FOURIER SERIES 2 Th scond lin givs th larg asymptotic of th function k (!), (6.8) k (!) 2! 2 (! 2 ) (! 2 2 ) (! 2 2 ) + o! 2 as! 2! but! 2 6 n2 for a positiv intgr n. In particular, whn!! n ar larg zros of th basic sin function S (!) on gts as n! (6.9) k (! n ) k 4n 2 ( )2 ( 2 ) 2 by (5.) (I.9) of [7]. With th aid of (6.6){(6.7) on can now rwrit (3.3) (3.32) in mor xplicit form, (6.) (6.) C 2 (cos!) k (!) + 2 2! ( ) 2 S 2 (cos!) k (!) 2 2! ( ) 2 ( 2 2 ) d C (!) S (!) ( 2 2 ) d C (!) S (!) : Ths basic intgrals ar, obviously, -xtnsions of th following lmntary intgrals (6.2) cos 2!x dx +! sin! cos! (6.3) sin 2!x dx! sin! cos! rspctivly. Whn! satiss th boundary condition (3.32) th last trms in th right sids of (6.) (6.) vanish w obtain th valus of th normalization constants in th orthogonality rlations (3.26){(3.28) in trms of th function k (!) dnd by (6.7). 7. Orthogonality Rlations for -Exponntial Functions Eulr's formula, (7.) i!x cos!x + i sin!x

22 22 JOAQUIN BUSTO AND SERGEI K. SUSLOV allows us to rwrit th orthogonality rlations for th trigonomtric functions (.6){ (.8) in a complx form, l xp i m x xp i n (7.2) 2l l l x dx mn whr (7.3) l mn Th -analog of Eulr's formula (7.) is (7.4) if m n if m 6 n: E (x i!) C (x!) + is (x!) whr E (x ) with i! is th -xponntial function introducd in [5] (s also [5] [24], w shall us th sam notations as in [24]) C (x!) S (x!) ar basic cosin sin functions dnd by (2.) (2.2), rspctivly. Our orthogonality rlations for th basic trigonomtric functions (3.26){(3.28) rsult in th following orthogonality proprty for th -xponntial function (7.5) 2k (! n ) E (cos i! m ) E (cos i! n ) d ( 2 2 mn ) whr! m! n!! 2! 3 :::!!! 2! 3 :::, ar nonngativ zros of th basic sin function S (!) arrangd in ascnding ordr of magnitud th normalization constants k (! n ) ar dnd by (6.7). A basic analog of (7.6) is (7.7) i!(x+y) cos! (x + y) + i sin! (x + y) E (x y i!) C (x y!) + is (x y!) s [5] [24]. Th gnral xponntial function on a -uadratic grid E (x y i!) has th following orthogonality proprty. Thorm 7.. (7.8) E (cos cos ' i! m ) E (cos cos ' i! n ) ( 2 2 ) d 2k (! n ) E (cos ' i! n ) E (cos ' i! n ) mn whr! m! n!! 2! 3 :::!!! 2! 3 :::, ar nonngativ zros of th basic sin function S (!) arrangd in ascnding ordr of magnitud th normalization constants k (! n ) ar dnd by (6.7).

23 -FOURIER SERIES 23 Proof. Using of th \addition" thorm for basic xponntial functions [24], (7.9) E (x y i!) E (x i!) E (y i!) th orthogonality rlation (7.5) on gts (7.8). In a similar fashion, w can stablish th following rsults. Thorm 7.2. (7.) C (cos cos '! m ) C (cos cos '! n ) d ( 2 2 ) if m 6 n k (! n ) C (cos ' cos '! n ) if m n (7.) S (cos cos '! m ) S (cos cos '! n ) d ( 2 2 ) if m 6 n k (! n ) C (cos ' cos '! n ) if m n (7.2) C (cos cos '! m ) S (cos cos '! n ) d ( 2 2 ) if m 6 n k (! n ) S (cos ' cos '! n ) if m n whr! m! n!! 2! 3 :::, ar positiv zros of th basic sin function S (!) arrangd in ascnding ordr of magnitud th normalization constants k (! n ) ar dnd by (6.7). Proof. Us th \addition" thorm for th basic trigonomtric functions [24] th orthogonality rlations (3.26){(3.28). 8. Basic Fourir Sris By analogy with (.2) w can now introduc a -vrsion of Fourir sris, (8.) f (cos ) a + X n (a n C (cos! n ) + b n S (cos! n ))

24 24 JOAQUIN BUSTO AND SERGEI K. SUSLOV whr!!! 2! 3 :::, ar nonngativ zros of th basic sin function S (!) arrangd in ascnding ordr of magnitud, (8.2) a 2k () f (cos ) ( 2 2 ) d (8.3) a n k (! n ) f (cos ) C (cos! n ) ( 2 2 ) d (8.4) b n k (! n ) f (cos ) S (cos! n ) ( 2 2 ) d: Th complx form of th basic Fourir sris (8.) is (8.5) with (8.6) c n f (cos ) 2k (! n ) X n c n E (cos i! n ) f (cos ) E (cos i! n ) ( 2 2 ) d whr! n!! 2! 3 :::!!! 2! 3 :::, ar nonngativ zros of th basic sin function S (!) arrangd in ascnding ordr of magnitud th normalization constants k (! n ) ar dnd by (6.7). Ths xprssions, of cours, mrly indicat how th cocints of our basic Fourir sris ar to b dtrmind on th hypothsis that th xpansion xists is uniformly convrgnt. W shall study th ustion of convrgnc of th sris (8.) (8.5) in th nxt sctions. Th -Fourir sris of f in ithr of th forms (8.) (8.5) will b dnotd in a usual mannr by S [f]. 9. Compltnss of th -Trigonomtric Systm Compltnss of th trigonomtric systm f inx g n on th intrval ( ) is on of th fundamntal facts in th thory of trigonomtric sris (s, for xampl, [], [8], [2], [9] [3]). In this sction w shall prov a similar proprty for th systm of basic trigonomtric function fe (x i! n )g, whr! n!! 2! 3 :::!!! 2! 3 :::, ar nonngativ zros of th basic sin function S (!)

25 -FOURIER SERIES 25 arrangd in ascnding ordr of magnitud. But rst w nd to discuss connctions btwn th basic trigonomtric functions th continuous -Hrmit polynomials. Th continuous -Hrmit polynomials, (9.) H n (cos j) hav two gnrating functions, (9.2) whn jrj < (9.3) X n X n nx k r n ( ) n H n (cos j) (s, for xampl, [7], [5], [24]). Lmma 9.. Th following functions (9.4) (9.5) ( ) n ( ) k ( ) nk i(n2k) (r i r i ) n2 4 ( ) n n H n (cos j) 2 2 E (cos ) (x ) 2 2 E (x ) s(!)! 2 2 S (!) ((x i!) (x i!)) (9.6) c(!)! 2 2 C (!) 2 ((x i!) + (x i!)) ar ntir functions in!, rspctivly, of ordr zro for all ral valus of x. Proof. Th gnrating function (9.3) givs a powr sris xpansion for th function (9.4), (9.7) with (9.8) (x ) X n h n h n (x) n2 4 h n n ( ) n H n (xj) : Th radius of convrgnc of this sris is innity, bcaus (9.9) R lim (jh nj) n n!! n n24 lim H n (xj) n! : ( ) n

26 26 JOAQUIN BUSTO AND SERGEI K. SUSLOV Thus, (x ) is an ntir function in. Th ordr of this ntir function is [9] n log n (9.) lim n! log jh n j n log n lim : n! log j n2 4 H n (xj) ( ) n j Functions (9.5) (9.6) ar just a sum or dirnc of two functions of typ (9.4), so thy ar also ntir functions of ordr zro. This provs th lmma. Th nxt stp is to stablish th following inualitis. Lmma 9.2. Lt cosh x cosh whr x cos,,. Thn (9.) j (cos )j (cosh jj) (9.2) (cosh jj) (cosh jj) if <. Proof. On can rwrit (9.) as [n2] X ( ) (9.3) H n (cos j) 2 n cos (n 2k) : ( ) k ( ) nk Thus, (9.4) k [n2] X jh n (cos j)j 2 Estimating both sids of (9.3) givs k ( ) n ( ) k ( ) nk H n (cosh j) : 2 2 E (cos ) X X n n n2 4 ( ) n n2 4 ( ) n jj n jh n (cos j)j jj n H n (cosh j) jj 2 2 E (cosh jj) cosh (n 2k) by (9.4) (9.3). This provs (9.). Th monotonicity proprty (9.2) follows from th monotonicity of th hyprbolic cosin function. It is clar that th systm fe (x i! n )g n is complt if th uivalnt systm f (x i! n )g n is closd.

27 -FOURIER SERIES 27 Suppos that th systm f (x i! n )g n is not closd on ( ). This mans that thr xists at last on function (x), not idntically zro, such that (9.5) (x) (x i! n ) (x) dx n 2 ::: whr (x) is th absolutly continuous masur in th orthogonality rlation (7.5). Thn, th function (9.6) f(!) (x) (x i! n ) (x) dx is an ntir function of ordr zro f(! n ) for all n 2 :::. Thus th study of closur amounts to th study of zros of a crtain ntir function. Suppos that (x) is intgrabl on ( ), (9.7) Thn (9.8) by (9.) (9.7). Considr th uotint (9.9) jf(!)j j(x)j (x) dx A < : j(x) (x i!)j (x) dx (cosh j!j) j(x)j (x) dx A (cosh j!j) g(!) f(!) s(!) of two ntir functions, f(!) s(!) dnd by (9.6) (9.5), rspstivly. Th functions f(!) s(!) hav th sam zros, so g(!) is an ntir function. Th ordr of this ntir function is zro bcaus both f(!) s(!) ar of ordr zro (s [9], Corollary of Thorm 2 on p. 24). Morovr, this function g(!) is boundd on a straight lin paralll to th imaginary axis. Indd, lt! + i. Using th sam argumnts as in Sction 5 on can s that (9.2) lim jj! s(i) ( jj) < : From this condition th inuality (9.8), it follows that th ntir function g(!) is boundd on th imaginary axis. But an ntir function of ordr zro boundd on a lin must b a constant (s Thorms 2{22 Corollary on pp. 49{5 of [9]). Thn, (9.2) f(!) c s(!)

28 28 JOAQUIN BUSTO AND SERGEI K. SUSLOV, thrfor, (9.22) jcj (x) (x i!) s(!) (x) dx (x i!) (x) s(!) (x) dx A E (cosh j!j) S(cosh j!j)! as j!j! <. Thus, f(!) is idntically zro th function (x) dos not xist. W hav stablishd th following thorm. Thorm 9.3. Th systm of th basic trigonomtric function fe (x i! n )g, whr n 2 :::!!! 2! 3 :::, ar nonngativ zros of th basic sin function S (!) arrangd in ascnding ordr of magnitud, is complt on ( ). As corollaris w hav th following rsults. Thorm 9.4. If f(x) g(x) hav th sam -Fourir sris, thn f g. Proof. Th -Fourir cocints of f g all vanish, so that f g. Thorm 9.5. If f(x) is continuous S [f], th -Fourir sris of function f, convrgs uniformly, thn its sum is f(x). Proof. Lt g(x) dnot th sum of S [f], th -Fourir sris in th right sid of (8.5). Thn th cocints of S [f] ar -Fourir cocints of g. Hnc, S [f] S [g], so that f g, f g bing continuous, f(x) g(x). Bssl's inuality for th -trigonomtric systm fe (x i! n )g n, whr!!! 2! 3 :::, ar nonngativ zros of th basic sin function S (!) arrangd in ascnding ordr of magnitud, taks th form (9.23) NX nn jc n j 2 jf(x)j 2 (x) dx providd f 2 L 2 ( ), which mans that jf(x)j 2 is intgrabl on ( ) with rspct to th wight function (x) in th orthogonality rlation (7.5). Hr c n ar th -Fourir cocints of f(x) dnd by (8.6). Whn N! w gt Parsval's formula (9.24) X n jc n j 2 jf(x)j 2 (x) dx du to th compltnss of th -trigonomtric systm fe (x i! n )g n th spac L 2 ( ) [], [8]. It follows that th -Fourir cocints c n tnd to zro if f 2 L 2 ( ).

29 -FOURIER SERIES 29. Bilinar Gnrating Function In this sction w shall driv th following bilinar gnrating rlation, (.) X n (r 2! 2 n 2 ) (! 2 n 2 ) k (! n ) E (cos i! n ) E (cos ' ir! n ) r2 2 2 (r i+i' r ii' r i'i r ii' ) for th basic xponntial functions. Hr as bfor! n!! 2! 3 :::!!! 2! 3 ::: ar nonngativ zros of th basic sin function S (!) arrangd in ascnding ordr of magnitud. W shall us this gnrating function for a furthr invstigation of th convrgnc of th basic Fourir sris (8.5) in th subsunt sction. Lt us stablish a conncting rlation of th form, (.2) ( 2 r 2 2 ) E (cos r) ( 2 2 ) ( r 2 ' ' ) 2 (r i+i' r ii' r i'i r ii' ) E (cos ' ) d' whr jrj <. On can asily s that if w could prov th uniform convrgnc in th variabl x cos of th sris in th lft sid of (.), than th intgral in (.2) givs th corrct valus of th basic Fourir cocints (s (8.5){(8.6)), which vris th gnrating rlation (.) by Thorm 9.5. So, on nds to giv a prov of (.2) rst. Th continuous -Hrmit polynomials hav th following bilinar gnrating function (th Poisson krnl), (.3) X n r n ( ) n H n (cos j) H n (cos 'j) (r 2 ) (r i+i' r ii' r i'i r ii' ) whr jrj <. Th orthogonality rlation for ths polynomials is (.4) H m (cos j) H n (cos j) d 2 ( ) n ( ) mn (s, for xampl, [7]). Exping E (cos ' ) in th right sid of (.2) in th uniformly convrgnt sris of th -Hrmit polynomials with th aid of (9.3), w

30 3 JOAQUIN BUSTO AND SERGEI K. SUSLOV gt (.5) 2 ( r 2 ' ' ) (r i+i' r ii' r i'i r ii' ) 2 2 E (cos ' ) d' X n 2 n2 4 ( ) n n ( r 2 ' ' ) (r i+i' r ii' r i'i r ii' ) H n (cos 'j) d': Th sris in (.3) convrgs uniformly whn jrj <. Thn, using (.4), (.6) 2 H n (cos 'j) ( r 2 ' ' ) (r i+i' r ii' r i'i r ii' ) d' r n H n (cos j) : From (.5), (.6), (9.3) w nally arriv at th conncting rlation (.2). Uniform convrgnc of th sris in (.) can b justid with th hlp of th inuality (9.) th corrsponding asymptotic xprssions. This provs (.) by Thorm 9.5. It is worth mntioning a fw spcial cass of (.). Whn r w obtain th following gnrating function, (.7) X n for E (x i! n ). If ' 2, on gts (.8) (! 2 n 2 ) k (! n ) E (cos i! n ) 2 2 X n (r 2! 2 n 2 ) (! 2 n 2 ) k (! n ) E (cos i! n ) r2 2 2 (r 2 r 2 2 ) : A trminating cas of this gnrating rlation appars whn r 2 intgr m 6, (.9) jmj X njmj (! 2 n! 2 m 2 ) (! 2 n 2 ) k (! n ) E (cos i! n )! 2 m for an

31 Hr m 2 3 :::. -FOURIER SERIES 3 r2 2 2 (! 2 m! 2 m 2 ) :. Mthod of Summation of Basic Fourir Sris According to Thorm 9.5, for a continuous function f(x) th basic Fourir sris S [f] convrgs to f(x) if it convrgs uniformly. In this sction w shall discuss anothr mthod of summation of basic Fourir sris. Lt f(x) b a boundd function that is continuous on ( ) lt S [f] b its -Fourir sris dnd by th right sid of (8.5). Rplac this sris by (.) whr S r [f] X n c n (r) E (cos i! n ) (.2) c n (r) (r2! n 2 2 ) (! n 2 2 ) f (cos ) E (cos ir! n ) 2k (! n ) providd that < r <. Comparing (.2) (8.6), (.3) ( 2 2 ) d lim c n (r) c n r! whr c n ar th rgular -Fourir cocints of f(x). Suppos that th sris S r [f] convrgs uniformly with rspct to th paramtr r whn < r <. Thn, (.4) lim S r [f] S [f] : r! On th othr h, from (.){(.2) on gts (.5) S r [f] X n 2k (! n ) (r 2! n 2 2 ) E (cos i! n ) (! n 2 2 ) f (cos ') E (cos ' ir! n ) ( ' ' ) ( 2 ' 2 ' ) d': Using th uniform convrgnc of th sris in th bilinar gnrating function (.), w nally obtain (.6) S r [f] 2 f (cos ') ( r 2 ' ' ) (r i+i' r ii' r i'i r ii' ) d':

32 32 JOAQUIN BUSTO AND SERGEI K. SUSLOV It has bn shown in [3] (s also [29]) that (.7) lim r! 2 f (cos ') ( r 2 ' ' ) (r i+i' r ii' r i'i r ii' ) d' f (cos ) for vry boundd function f (cos ) that is continuous on < <. As a rsult w hav provd th following thorm. Thorm.. Lt f(x) b a boundd function that is continuous on ( ) lt S r [f] b th sris dnd by (.){(.2). If S r [f] convrgs uniformly with rspct to th paramtr r whn < r <, thn lim r! S r [f] S [f] f(x). 2. Rlation Btwn -Trigonomtric Systm -Lgndr Polynomials Th trigonomtric systm f inx g n th systm of th Lgndr polynomials fp m (x)g ar two complt systms in m L2 ( ). Th corrsponding unitary transformation btwn ths two orthogonal basiss its invrs ar (2.) (2.2) inx P m (x) 2 2 X i n m (m + 2) J m+2 (n) P m (x) m X n 2 (i) m J m+2 (n) inx 2n rspctivly. Rlation (2.) is a spcial cas of a mor gnral xpansion, 2 X (2.3) irx () i m ( + m) J +m (r) C m (x) r m whr C m (x) ar ultrasphrical polynomials J +m (r) ar Bssl functions [27]. Expansion (2.2) is th Fourir sris of th Lgndr polynomials on ( ). Orthogonality proprtis of th trigonomtric systm Lgndr polynomials lad to th orthogonality rlations, (2.4) (2.5) X m X n m + 2 n m + 2 n J m+2 (n) J m+2 (l) nl J m+2 (n) J p+2 (n) mp for th corrsponding Bssl functions. Th basic trigonomtric systm fe (x i! n )g n th systm of th continuous -ultrasphrical polynomials fc m (x j )g with m 2, which ar th basic analogs of th Lgndr polynomials, ar two complt orthogonal systms in L 2 ( ), whr is th wight function in th orthogonality rlation (7.5). Thrfor, thr xists a -vrsion of th unitary transformation (2.){(2.2).

33 -FOURIER SERIES 33 Ismail hang [5] hav found th following -analog of (2.3), (2.6) E (x i!) ( )! (! 2 2 ) ( ) X m i m +m m2 4 J (2) +m (2! ) C m (x j ) whr J (2) +m (2! ) is Jackson's -Bssl function (s, for xampl, [7]). Spcial cas 2 givs th basic analog of th xpansion (2.), (2.7) E (x i! n ) ( )! 2 n (! 2 n 2 ) ( 2 ) X m i m m+2 m2 4 J (2) m+2 (2! n ) C m x 2 whr! n! n!!! 2! 3 :::, ar nonngativ zros of th basic sin function S (!) arrangd in ascnding ordr of magnitud. On th othr h, th continuous -ultrasphrical polynomials C m x 2 can b xpd in th -Fourir sris as (2.8) C m x 2 2 Indd, by (8.5){(8.6), (2.9) whr (2.) c n ( ) X n C m x 2 2k (! n ) (i) m m2 4! n 2 J (2) k (! n ) (! n 2 2 ) (2! m+2 n ) E (x i! n ) : X n c n E (x i! n ) C m cos 2 E (cos i! n ) d: ( 2 2 ) Using (2.7), whr th sris on th right convrg uniformly in x for any!, th orthogonality rlation (2.) C m cos 2 Cp cos 2 d ( 2 2 )

34 34 JOAQUIN BUSTO AND SERGEI K. SUSLOV 2 (s, for xampl, [7]), on gts 2 2 ( ) 2 m+2 mp (2.2) or, (2.3) c n c n 2 ( ) (i) m m ( ) 2 2 ' m ( ) 2! 2 n k(! n ) (! 2 n 2 ) J (2) m+2 (2! n ) (i) m m2 4! n m (! n 2 2 ) ( 2 ) m+ k (! n ) (! n 2 2 ) m+52 2m+3 (i) m m2 4! m n ( 2 ) m+ k (! n ) 2! 2 n " m+32 ( 2m+3 2 )! 2 n m+32 2m! 2 n 2 (! 2 n 2! 2 n 2 ) m+32 2 ' m+2 2m 2! 2 n (2.4) +! 2 n m+52 2m! n 2 2 ( 2m+3 2 ) (! n 2 2! n 2 2 ) # 2 ' m+52 32m 3 2! 2 n by (5.3) (III.32) of [7], rspctivly. Th last uation givs th larg!- asymptotic of th basic Fourir cocints. With th aid of (5.), (6.9), (I.9) of [7], w nally obtain (2.5) as n!, whr D is som constant. jc n j s D n2! Thrfor, th sris on th right sid of (2.8) convrgs uniformly w hav stablishd th xpansion of th -Lgndr polynomials C m x 2 in trms of th basic trigonomtric functions E (x i! n ) du to Thorm 9.5. Rlations (2.7){(2.8) dn th unitary oprator acting in L 2 ( ) [2]. Orthogonality rlations of th matrix of this oprator lad to th following orthogonality proprtis X (2.6) m m+2! n k(! n ) (! 2 n 2 ) 2 m2 2

35 -FOURIER SERIES 35 (2.7) J (2) m+2 (2! n ) J (2) m+2 (2! l ) nl X n m+2! n k(! n ) (! 2 n 2 ) 2 m2 2 J (2) m+2 (2! n ) J (2) p+2 (2! n ) mp for th corrsponding Jackson's -Bssl function. Ths rlations ar, clarly, - analogs of (2.4){(2.5). 3. Som Basic Trigonomtric Idntitis On of th most important formulas for th trigonomtric functions is th main trigonomtric idntity, (3.) cos 2!x + sin 2!x : It follows from th Pythagoran Thorm or from th addition formulas for th trigonomtric functions, but on can also prov this idntity on th bas of th diffrntial uation. Th functions cos!x sin!x ar two solutions of (.9) corrsponding to th sam ignvalu!. Thrfor, (3.2) or (3.3) d dx [W (cos!x sin!x)] cos 2!x + sin 2!x constant: Substituting x, on vris (3.). On can xtnd this considration to th cas of th basic trigonomtric functions. Considr uation (3.8) with u(z) C (x(z)!), v(z) S (x(z)!), (z), (3.4) whr (3.5) [W (u(z) v(z))] W (u v) W (C (x!) S (x!)) 24! [C (x(z)!) C (x(z 2)!) + S (x(z)!) S (x(z 2)!)] is th analog of th Wronskian (3.9) w also usd (2.2){(2.3). On can asily s that W (u v) hr is a doubly priodic function in z without pols in th rctangl on th Figur. Thrfor, this function is just a constant by Liouvill's thorm, C (x (z)) C (x (z 2)) + S (x (z)) S (x (z 2)) C: Th valu of this constant C can b found by choosing x, which givs C (!2 2 ) 2 (! 2 2 ) 2

36 36 JOAQUIN BUSTO AND SERGEI K. SUSLOV 2 ' 2 ' 2! 2 2! 2 (!2 2 ) 2 (! 2 2 ) 2 ' 2! 2 (!2 2 ) (! 2 2 ) by th -binomial thorm. As a rsult on gts (3.6) C (cos!) C (cos ( + i log 2)!) + S (cos!) S (cos ( + i log 2)!) (!2 2 ) (! 2 2 ) as a -xtnsion of th main idntity (3.). Th spcial cas z 4, whn x(4), of (3.6) has th simplst form (3.7) C 2 (!) + S 2 (!) (!2 2 ) (! 2 2 ) : Our idntity (3.6) can also b drivd as a spcial cas of th \addition" thorm for th basic trigonomtric functions stablishd in [24]. In a similar fashion, w can nd an analog of th idntity (3.8) cos 2! (x + y) + sin 2! (x + y) considring mor gnral basic sin cosin functions, C (x y!) S (x y!), as two solutions of uation (2.). Th rsult is (3.9) C (cos cos '!) C (cos ( + i log 2) cos '!) + S (cos cos '!) S (cos ( + i log 2) cos '!) (!2 2 ) (! 2 2 ) C 2 (cos '!) + S 2 (cos '!) (!2 2 ) (! 2 2 ) C (cos ' cos '!) : W hav usd (6.2) hr. This idntity can also b vrid with th aid of th \addition" thorms for th basic trigonomtric functions. Idntity (3.7) givs th valus of th basic cosin function C (!) at th zros of th basic sin function S (!), (3.) C (! n ) () n s (! 2 n 2 ) (! 2 n 2 )

37 -FOURIER SERIES 37 vic vrsa, (3.) S ( $ n ) () n s ($ 2 n 2 ) ($ 2 n 2 ) with th aid of Thorm Exampl Lt us considr a priodic function p (x) which is dnd in th intrval ( ) by p (x) x. Its Fourir cocints ar Thrfor, (4.) c c n 2 x x inx dx ()n n 6 : in 2 Th spcial cas m of (2.8), (4.2) X n X n () n n C x 2 i 2 X n inx () n sin nx n : ( ) 4! 2 n k (! n ) (! 2 n 2 ) J (2) 32 (2! n ) E (x i! n ) givs us a possibility to stablish th -analog of (4.). Lt us rst simplify th right sid of (4.2). Using th thr-trm rcurrnc rlation for th -Bssl functions (s Exrcis.25 of [7]) (5.32){(5.33), on gts (4.3) J (2) 32 (2! n ) 2 J (2) 2 (2! n ) On th othr h, (4.4) 2 ( ) (! 2 n 2 ) (! n ) 2 C (! n ) : C x 2 2 x: 2 +

38 38 JOAQUIN BUSTO AND SERGEI K. SUSLOV Combining (4.2){(4.4) (3.), w nally obtain, (4.5) x 2 2 ( ) 2 X n 2 2 ( ) 2 X n s () n (! n 2 2 ) i k (! n )! n (! n 2 2 ) E (x i! n ) s () n (! n 2 2 ) k (! n )! n (! n 2 2 ) Ths uations ar, clarly, -analogs of (4.). S (x! n ) : 5. Miscllanous Rsults Undr crtain rstrictions a function f(z) analytic in th ntir complx plan having zros at th points a a 2 a 3 ::: (ths ar th only zros of f(z)), whr lim n! ja n j is innit, can b rprsntd as an innit product (5.) f(z) f() zf ()f () Y zan zan n s, for xampl, [28], [9]. Considr th ntir function f(!)! 2 2 S(!) (5.2)! which has simpl ral zros at!! n by Thorms 5.{5.3. In this cas f() f(!) 2 2 f(!) f() + 2 f ()! 2 + ::: f () Y n!!n!!n +!!n!!n : As a rsult w arriv at th innit product rprsntation for th basic sin function, (5.3) S(!)! 2 (! 2 2 )

39 -FOURIER SERIES 39 Y!!n!!n +!!n!!n n 2! (! 2 2 ) Y n!2 In a similar mannr, on can obtain an innit product rprsntation for th basic cosin function, (5.4) C(!) (! 2 2 )! 2 n : Y!$n!$n +!$n!$n n (! 2 2 ) Y n!2 Euations (3.){(3.) (5.3){(5.4) rsult in th following rlations, (5.5) (5.6) () n (! 2 m ) () n ($ 2 m ) $ 2 n Y n Y n :!2 m $ 2 n $2 m! 2 n btwn th zros of th basic sin S(!) basic cosin C(!) functions. 6. Appndix: Estimat of Numbr of ros of S(!) In this sction w giv an stimat for numbr of zros of th basic sin function S(!) on th basis of Jnsn's thorm (s, for xampl, [6] [9]). W shall apply th mthod proposd by Mourad Ismail at th lvl of th third Jackson - Bssl functions [](s also [3] for an xtnsion of his ida to -Bssl functions on a -uadratic grid). Lt us considr th ntir function f(!) dnd in (5.2) again lt n f (r) b th numbr of of zros of f(!) in th circl j!j < r. Considr also circls of radius R R n { n, 4 { < 34 with n 2 3 ::: in th complx!-plan. Sinc n f (r) is nondcrasing with r on can writ (6.) if R n r R n+,, thrfor, (6.2) n f (R n ) Rn+ Rn n f (R n ) n f (r) n f (R n+ ) dr Rn+ r n f (r) Rn r dr n f (R n+ ) Rn+ Rn dr r :

40 4 JOAQUIN BUSTO AND SERGEI K. SUSLOV But, nally, on gts (6.3) Rn+ Rn log n f (R n ) dr r log r Rn+ Rn+ Rn n f (r) r Rn log dr log n f (R n+ ) : In th proof of Thorm 5. w hav stablishd th fact that for sucintly larg n thr ar at last two roots of f(!) btwn th circls j!j R n j!j R n+. Thus, for sucintly larg n th inuality (6.3) should rally hav on of th following forms (6.4) or (6.5) log n f (R n ) log n f (R n ) < Rn+ Rn Rn+ Rn n f (r) r n f (r) r dr < log n f (R n+ ) dr log n f (R n+ ) : Our nxt stp is to stimat th intgral in (6.4){(6.5). By Jnsn's thorm [6], [9] (6.6) Rn+ Rn n f (r) r dr Rn+ 2 2 n f (r) dr r log Rn f { n i# f ({ n i# ) For larg valus of n in viw of (5.5), f { n i# f ({ n i# ) ( 32 { 2 2n # 2 ) 32 { 2 2n2 # log whr { 2 2. Thrfor, (6.7) Rn+ Rn as n!. From (6.3) (6.7), + f { n i# f ({ n i# ) n f (r) r log log 2n 32 { 2 2n2 # 2 2n log + log n f (r) dr r dr 2n log + log + o () n n f (R n ) 2n + d#: log log 2n

41 , thrfor, n f (R n ) (6.8) lim n! 2n On th othr h, from (6.4){(6.5), (6.9) or (6.) which givs -FOURIER SERIES 4 : n f (R n ) 2n + log log < n f (R n+ ) n f (R n ) < 2n + log log n f (R n+ ) n f (R n+ ) n f (R n ) < 2n log log 2n 2 + log log 4 Thus, w hav stablishd that (6.) n f (R n+ ) n f (R n ) < 4 as n!. Du to th symmtry f (!) f (!) th last inuality implis that thr is only on positiv root of S(!) btwn th tst points! n dnd by (5.6) for larg valus of n. 7. Acknowldgmnts W wish to thank Dick Asky, Mourad Ismail, John McDonald, Mizan Rahman for valuabl discussions commnts. On of us (S. S.) gratfully acknowldg th hospitality of th Dpartmnt of Mathmatics at Arizona Stat Univrsity wr this work was don. Rfrncs [] N. I. Akhizr, Thory of Approximation, Frdrick Ungar Publishing Co., Nw York, 956. [2] N. I. Akhizr I. M. Glazman, Thory of Linar Oprators in Hilbrt Spac, Dovr, Nw York, 993. [3] R. A. Asky, M. Rahman, S. K. Suslov, On a gnral -Fourir transformation with nonsymmtric krnls, J. Comp. Appl. Math. 68 (996), 25{55. [4] R. A. Asky J. A. Wilson, Som basic hyprgomtric orthogonal polynomials that gnraliz Jacobi polynomials, Mmoirs Amr. Math. Soc., Numbr 39 (985). [5] N. M. Atakishiyv S. K. Suslov, Dirnc hyprgomtric functions, in: \Progrss in Approximation Thory: An Intrnational Prspctiv", ds. A. A. Gonchar E. B. Sa, Springr Sris in Computational Mathmatics, Vol. 9, Springr{Vrlag, 992, pp. {35. [6] R. P. Boas, Entir Functions, Acadmic Prss, Nw York, 954. [7] G. Gaspr M. Rahman, Basic Hyprgomtric Sris, Cambridg Univrsity Prss, Cambridg,99. [8] Y. Chn, M. E. H. Ismail, K. A. Muttalib, Asymptotics of basic Bssl functions -Lagurr polynomials, J. Comp. Appl. Math. 54 (994), 263{272. [9] M. E. H. Ismail, Th basic Bssl functions polynomials, SIAM J. Math. Anal. 2 (98), 454{468. [] M. E. H. Ismail, Th zros of basic Bssl functions, th functions J+ax (x), associatd orthogonal polynomials, J. Math. Anal. Appl. 86 (982), {9. [] M. E. H. Ismail, Som proprtis of Jackson's third -Bssl functions, to appar.

42 42 JOAQUIN BUSTO AND SERGEI K. SUSLOV [2] M. E. H. Ismail, D. R. Masson, S. K. Suslov, Th -Bssl functions on a -uadratic grid, to appar. [3] M. E. H. Ismail, D. R. Masson, S. K. Suslov, Proprtis of a -analogu of Bssl functions, to appar. [4] M. E. H. Ismail M.E. Muldoon, On th variation with rspct to a paramtr of zros of Bssl -Bssl functions, J. Math. Anal. Appl. 35 (988), 87{27. [5] M. E. H. Ismail R. hang, Diagonalization of crtain intgral oprators, Advancs in Math. 8 (994), {33. [6] M. E. H. Ismail, M. Rahman, R. hang, Diagonalization of crtain intgral oprators II, J. Comp. Appl. Math. 68 (996), 63{96. [7] R. Kokok R. F. Swarttow, Th Asky schm of hyprgomtric orthogonal polynomials its -analogus, Rport 94{5, Dlft Univrsity of Tchnology, 994. [8] A. N. Kolmogorov S. V. Fomin, Introductory Ral Analysis, Dovr, Nw York, 97. [9] B. Ya. Lvin, Distribution of ros of Entir Functions, Translations of Mathmatical Monographs, Vol. 5, Amr. Math. Soc., Providnc, Rhod Isl, 98. [2] N. Lvinson, Gap Dnsity Thorms, Amr. Math. Soc. Collo. Publ., Vol. 36, Nw York, 94. [2] A. F. Nikiforov, S. K. Suslov, V. B. Uvarov, Classical Orthogonal Polynomials of a Discrt Variabl, Nauka, Moscow, 985 [in Russian] English translation, Springr{Vrlag, Brlin, 99. [22] M. Rahman, An intgral rprsntation som transformation proprtis of -Bssl functions, J. Math. Anal. Appl. 25 (987), 58{7. [23] S. K. Suslov, Th thory of dirnc analogus of spcial functions of hyprgomtric typ, Russian Math. Survys 44 (989), 227{278. [24] S. K. Suslov, \Addition" thorms for som -xponntial -trigonomtric functions, Mthods Applications of Analysis, to appar. [25] S. K. Suslov, Som orthogonal vry-wll-poisd 8 ' 7 -functions, J. Phys. A: Math. Gn., submittd as a Lttr. [26] S. K. Suslov, Som orthogonal vry-wll-poisd 8 ' 7 -functions that gnraliz th Asky{Wilson polynomials, undr prparation. [27] G. N. Watson, A Tratis on th Thory of Bssl Functions, scond dition, Cambridg Univrsity Prss, Cambridg, 944. [28] E. T. Whittakr G. N. Watson, A Cours of Modrn Analysis, fourth dition, Cambridg Univrsity Prss, Cambridg, 952. [29] N. Winr, Th Fourir Intgral Crtain of Its Applications, Cambridg Univrsity Prss, Cambridg, 933 Dovr dition publishd in 948. [3] A. ygmund, Trigonomtric Sris, scond dition, Cambridg Univrsity Prss, Cambridg, 968. Joauin Bustoz, Dpartmnt of Mathmatics, Arizona Stat Univrsity, Tmp, Arizona , U.S.A. addrss: bustoz@math.la.asu.du Srgi K. Suslov, Kurchatov Institut, Moscow, 2382, Russia Currnt addrss: Dpartmnt of Mathmatics, Arizona Stat Univrsity, Tmp, Arizona , U.S.A. addrss: suslov@math.la.asu.du