ON CONVERGENCE OF BASIC HYPERGEOMETRIC SERIES. 1. Introduction Basic hypergeometric series (cf. [GR]) with the base q is defined by

Transcription

1 ON CONVERGENCE OF BASIC HYPERGEOMETRIC SERIES TOSHIO OSHIMA Abstract. We examie the covergece of q-hypergeometric series whe q =. We give a coditio so that the radius of the covergece is positive ad get the radius. We also show that the umbers q with the positive radius of the covergece are desely distributed i the uit circle of the complex plae of q ad so are those with the radius.. Itroductio Basic hypergeometric series (cf. [GR]) with the base q is defied by [ ] a, a 2,..., a r (a ; q) (a 2 ; q) (a r ; q) ( rϕ s ; q, z = ( ) ) q ( ) 2 s+ rz, b,..., b s (q; q) (b ; q) (b s ; q) where = (a; q) = ( aq j ) j= is the q-pochhammer symbol. Here a,..., a r, b,..., b s ad q are complex parameters. I this paper we always assume () a i q ad b j q (i =,..., r, j =,..., s, =,, 2,...) so that the factors (a i ; q) ad (b j ; q) i the terms of the series are ever zero. Let v be the terms of the series r ϕ s which cotai z. The we have v + v = ( a q )( a 2 q ) ( a r q ) ( q + )( b q ) ( b s q ) ( q ) +s r z = (a q )(a 2 q ) (a r q ) z ( q )(b q ) (b s q ) q. If < q <, the radius of covergece of the series r ϕ s equals if r s ad equals if r = s +. If q > ad (2) a a r b b s, the radius of covergece of the series equals (3) b b 2 b s q a a 2 a r. I this paper we discuss the covergece of the series whe q =. The covergece of 2 ϕ is assumed i [OS] but it is a subtle problem depedig o the base (4) q = e 2πiθ. Key words ad phrases. Basic hypergeometric series, Diophatie approximatio, Distributio modulo oe. 2 Mathematics Subject Classificatio. Primary 33D5; Secodary 4A5, J7. Supported by Grat-i-Aid for Scietific Researches (B), No , Japa Society of Promotio of Sciece.

2 2 TOSHIO OSHIMA We assume that θ is ot a ratioal umber, amely, θ R \ Q so that (q; q) ever vaish ad we have the followig theorem. Theorem. Retai the otatio above ad assume the coditios () ad (2). i) Assume that there exists a positive umber C such that θ k m > C ( k Z, m =, 2, 3,...). The we have ( (5) lim e 2πiθ ; e 2πiθ) =. Suppose moreover that every parameter a i or b j has a absolute value differet from or equals e 2πiα q β with suitable ratioal umbers α ad β which may deped o a i ad b j. The the radius of covergece of the series r ϕ s equals (6) ii) I geeral, we have max{ b, } max{ b s, } max{ a, } max{ a r, }. (7) lim (e 2πiθ ; e 2πiθ ) The set of irratioal real umbers θ satisfyig ( θ R \ Q). (8) lim (e 2πiθ ; e 2πiθ ) = is dese i R ad ucoutable. If θ satisfies (8) ad the absolute value of ay parameter a i or b j is ot, the radius of covergece of the series r ϕ s equals. Note that it is kow that a irratioal umber θ satisfies the assumptio i Theorem i) if ad oly if the positive itegers appearig i its expasio of cotiued fractio are bouded ad hece the set of real umbers θ satisfyig it is ucoutable ad dese i R. Suppose θ R \ Q satisfies the assumptio i Theorem i). The (9) k θ k 2 k m m = k θ m (k 2 m + k ) k m > C k m 2 m2 m for itegers k, k, k 2, m, m, with k mm ad therefore the umber r θ + r 2 with r Q \ {} ad r 2 Q also satisfies the assumptio. Example 2. The estimate () 2 k > m 3 ( k Z, m =, 2, 3,...) shows that the real umber θ = 2 + r with r Q satisfies the assumptio i Theorem i). We will prove (). We assume the existece of itegers k ad m satisfyig m ad 2 k m 3m. The we may moreover assume m 2 ad therefore 2 2 k 2 = ( 2m k )( 2m + k ) = m2( k ) ( m) 2 m 2 k 2 2 ( ) =.975 <, which leads a cotradictio. We will show Theorem i) i 4 ad Theorem ii) i 5.

3 ON CONVERGENCE OF BASIC HYPERGEOMETRIC SERIES 3 2. Prelimiary results First we review the followig theorem which claims that kθ mod Z for k =, 2,... are uiformly distributed o R/Z. Theorem 3 (Bohl, Sierpiński ad Weil). Let f(x) be a periodic fuctio o R with period. If f(x) is itegrable i the sese of Riema, the () lim f(kθ) = f(x) dx ( θ R \ Q). k= k= This theorem is proved by approximatig f(x) by a fiite Fourier series (cf. [AA]) sice the theorem is directly proved if f(x) is a fiite Fourier series with the fact e 2πimkθ ( ) = e2πimθ e 2πimθ e 2πimθ (m ). We also prepare the followig itegral formula. log { re 2πix ( r ), (2) dx = log r (r ). The series log( z) z = + z 2 + z2 3 + coverges whe z < ad therefore log( z) = dz = 2πi log( re 2πix ) dx ( r <, z = 2πix). z z =r by Cauchy s itegral formula. Sice Re log( re 2πix ) dx = Re log( re 2πix ) dx = we have (2) whe r <. Moreover the relatio log re 2πix dx, log re 2πix = log r + log r e 2πix = log r + log r e 2πix assures (2) whe r >. Note that the expasio e σ = σ( + σ 2! + σ2 3! + ) assures eσ σ 2 whe σ <. Hece if r =, the improper itegratio i (2) coverges because (3) log e 2πiz > log πz for < 2πz < ad we obtai (2) by takig the limit r (cf. [Ah, 5.3.5]). 3. A lemma We prepare a lemma to prove Theorem i). Lemma 4. Let f(x) be a periodic fuctio o R with period. Suppose that f(x) is cotiuous o [, ] except for fiite poits c,..., c p [, ). Suppose there exist r j Q for j =,..., p such that (4) c j r j θ Q ad (r j + k)θ c j Z for k =, 2,.... Suppose moreover that there exist a positive umber ϵ ad a cotiuous fuctio h(t) o (, ] such that (5) f(x) < h( x c j ) for < x c j < ϵ, h(t) dt < ad h(t ) h(t 2 ) if < t < t 2. If θ R \ Q satisfies the assumptio i Theorem i), the () is valid. Here we ote that the coditio (5) assures that the improper itegral i () coverges.

4 4 TOSHIO OSHIMA Proof. Put ad J(j,, ϵ) = { k k, mi m Z { kθ c j m } < ϵ } I ϵ = { x [, ] mi m Z x c j m ϵ for j =,..., p }. The Theorem 3 shows that lim k J(,,ϵ) J(p,m,ϵ) k ad therefore we have oly to show (6) lim lim ϵ + k J(j,,ϵ) f(kθ) = f(x) dx I ϵ f(kθ) = to get this lemma. Fix j. Sice Theorem 3 shows that lim #J(j,, ϵ) = 2ϵ, there exists a positive iteger N ϵ such that #J(j,, ϵ) 3ϵ ( N ϵ ). Put J(j, ϵ, ) = {k, k 2,..., k L } with L = #J(j, ϵ, ) so that mi k νθ c j m mi k ν θ c j m if ν < ν L. m Z m Z Note that c j = k m θ + k 2 with itegers k, k 2, m,. I view of (9), we have kθ k θ k 2 m m = m k k θ k 2 + m m > C m 2 mk k for k =, 2, 3,... satisfyig mk k. If is a positive iteger, the assumptio implies k 2 Z. Hece replacig C by a small positive umber if ecessary, we may assume mi kθ c j m > C (k =, 2,..., j =,,..., p), m Z k where we put c =. I particular, we have mi kθ m Z k θ m > C k k C ( k < k ). Thus we have ad k m mi k νθ c j m > Cν m Z 2 f(k ν θ) < h ( ) Cν 2 Hece if N ϵ, we have k J(j,,ϵ) f(kθ) = ( ν L) ( ν L < 3ϵ). L f(k ν θ) ν= [3ϵ] [3ϵ] ν= h( Cν 2 ) h ( 6ϵ ) Cx 2 C 2 dx h(t) dt, C

5 ON CONVERGENCE OF BASIC HYPERGEOMETRIC SERIES 5 which implies (6). Here [3ϵ] deotes the largest iteger satisfyig [3ϵ] 3ϵ. Problem 5. Let f(x) be a fuctio satisfyig the assumptio i Lemma 4. Is the equality () for θ R valid almost everywhere i the sese of Lebesgue measure? Here we may assume p =, c = ad h(t) = log t for our problem. Is it also valid without assumig the coditio (4)? 4. Estimate I Let a = re 2πiτ ad q = e 2πiθ with τ, θ R ad r >. The ( (7) (a; q) = exp log ) re 2πi(kθ+τ). k= If r, Theorem 3 ad (2) imply (8) lim (a; q) = max{ a, } ( a =, q = e 2πiθ with θ R \ Q). Assume r =. Sice log e 2πikθ ad lim mi m Z { kθ m }<ϵ k mi m Z { kθ m } ϵ k log e 2πikθ = ϵ ϵ log e 2πix dx for ay small positive umber ϵ, we have (7) i view of (2) ad (7). Now assume moreover that θ satisfies the assumptio i Theorem i). The Lemma 4 with f(x) = log e 2πix ad h(t) = log πt (cf. (3)) proves (5). Suppose τ = k m θ + k 2 with itegers k, k 2, m, with m >, > ad ( ) k + k θ + k 2 / Z (k =, 2, 3,...) m correspodig to (), (2) ad (4). Lemma 4 with f(x) = log e 2πi(x+( k m )θ), h(t) = log πt ad c j = ( k m )θ k 2 implies (9) lim ( e 2πiτ ; e 2πiθ) =. Thus we have Theorem i) by the estimates (8), (5) ad (9). 5. Estimate II Defie a series of rapidly icreasig positive itegers {a } by (2) a = 2, a + = k + a a! (k + = 2 or 3, =, 2, 3,...) ad put θ = ad q = e 2πiθ. The θ Q ad we have a (2) = mia θ m < m Z e 2πia θ 2π a!, k=+ a q j π 2a a!, j= a < a k a!, lim a a j= q j =

6 6 TOSHIO OSHIMA ad (8) i Theorem ii). We may choose k {2, 3} for =, 2,..., we get ucoutably may θ s. Moreover if we put θ = = a + r so that there exists a positive iteger N satisfyig rn Z, the θ also satisfies (2) for N ad hece θ satisfies (8). The remaiig claim i Theorem ii) is clear from (8). Refereces [Ah] L. V. Ahlfors, Complex Aalysis, Third Editio, McGraw-Hill, 979. [AA] V. I. Arold ad A. Avez, Problèmes Ergodiques de la Mécaique Classiqe, Gauthier-Villars, 967. [GR] G. Gasper ad M. Rahma, Basic Hypergeometric Series, Secod Editio, Ecyclopedia of Mathematics ad its Applicatios 96, Cambridge Uiversity Press, 24. [OS] S. Odake ad R. Sasaki, Reflectioless potetials for differece Schrödiger equatios, J. Phys. A: Math. Theor. 48(25) 524 (2pp). Faculty of Sciece, Josai Uiversity, - Keyakidai, Sakado, Saitama , Japa address: t-oshima@josai.ac.jp