The roots of The Third Jackson q-bessel Function

Transcription

1 The roots of The Third Jacson q-bessel Function L. D. Abreu Departamento de Matemática Universidade de Coimbra Coimbra, Portugal J. Bustoz Department of Mathematics, Arizona State University Tempe, Arizona J. L. Cardoso Departamento de Matemática, Universidade de Tras-os-Montes e Alto Douro, Vila Real, Portugal May 8, Abstract We derive analytic bounds on the roots of the third Jacson Function. This bounds prove a conjecture of M. E. H. Ismail concerning the asymptotic behaviour of the roots. 1 Introduction. There are three nown q-analogs of classical Bessel functions [5,6] that are due to Jacson [7]. Following the notation of M. Ismail [5,6] these are designated by J υ () (z; q), = 1,, 3. The parameter q is taen to satisfy < q < 1. The third Jacson q-bessel function J v (3) (z; q) is defined as 1

2 [ J v (3) (z; q) := (qv+1 ; q) z v 1Φ 1 ] q v+1 ; q, qz This function is also nown as the Hahn-Exton q-bessel function [9,1 ]. The notation 1 Φ 1 in equation (1.1) is the standard in use for q-hypergeometric series [4]. J ν (3) (z; q) satisfies a linear q-difference equation and it is nown that J v (3) (z; q) has an infinite number of simple real zeros [9]. In this paper we will give lower and upper bounds for these zeros. The roots of these functions are of interest for several reasons. Firstly, it is intrinsically interesting to determine information about the roots of a function such as (1.1), which is an entire function of order zero. Also, the roots of J υ () (z; q) and J υ (3) (z; q) figure prominently in expansions in terms of q-fourier series [,3]. Lastly, if we denote the roots of J υ (3) (z; q) by j n,υ (3) then the mass points of the orthogonality measure for a q-analog of Lommel polynomials are located at the ponts 1/j n,υ. (3) Furthermore, although the functions defined in (1) are of a simpler character than the remaining Jacson q-bessel functions, it is hoped that the results given here for J υ (3) (z; q) may be extended in the future to J υ () (z; q), = 1,. The Roots of J υ (3) (z; q). We will prove two lemmas stating the existence of an odd number of roots in a certain interval and then we will prove that J v (3) (z; q) has only one root in each such interval. First we will apply the following transformation to (1): [ ] [ ] (c; q) 1 Φ 1 c ; q, z = (z; q) 1 Φ 1 z ; q, c This produces J v (3) (z; q) = (qz ; q) z v 1Φ 1 This last relation in series form gives υ (z; q) = zv [ ] qz ; q, q v+1 ( 1) (qz ; q) q (+v+1) (qz ; q) (q; q) = (1)

3 = zv = ( 1) (q+1 z ; q) (q; q) q (+v+1) () This representation will be critical in the proof of the next two lemmas. [ ] Lemma 1 If q v+1 < (1 q) then sgn J v (3) (q m ; q) = ( 1) m, m = 1,,... Proof: Set z = q m in (.1) to obtain q mv v (q m ; q) = ( 1) (q m++1 ; q) (q; q) = q (+v+1). Now observing that (q m++1 ; q) = if < m, the series on the right side of this last equality can be written as where ( 1) (q m++1 ; q) q (+v+1). (q; q) =m Setting j = m in this last series yields: (q; q) m J q mυ υ (3) (q m ; q) = ( 1) m ( 1) j A j j= A j = (qj+1 ; q) (q; q) j+m q (j+m)(j+m+v+1). Now we prove that A j+1 < A j.a calculation shows that A j+1 < A j is equivalent to q m+j+v+1 < (1 q m+j+1 )(1 q j+1 ). But the left hand side of this inequality is decreasing in m and j, while the right hand side of the inequality is increasing in m and j. So we only need to verify the case j = m =, that is q v+1 < (1 q), but this is the hypothesis of the Lemma. Clearly, since A j+1 < A j, then sgn( 1) m j= ( 1) j A j = ( 1) m. Lemma 1 states that there exist an odd number of roots in the interval (q m + 1, q m ). The next lemma will refine this statement. 3

4 Lemma Let q v+1 < (1 q) and define α (v) m (q) = log(1 q m+v 1 q m ). Then [ ] log q sgn (q m + α (v) m (q) ; q) = ( 1) m 1, m = 1,,... v Proof: First observe that the function α (v) m (q) is well defined, because if q v+1 < (1 q) then q v+1 < (1 q) and so, for positive integer m, q m+v < (1 q m ) 1 qm+v >. Being defined, it is clear that is positive, because 1 q m 1 qm+v < 1 holds for any q (, 1). 1 q m Observe also that α (v) m (q) < 1 log(1 qm+v ) > log q q m+v < 1 q m (1 q)(1 q m ), which is true if q v+1 < (1 q). So, we have < α (v) m (q) < 1. Now, set z = q m + α (v) m (q) in (). The substitution gives q mv + vα(v) m (q) v (q m + α (v) m (q) ; q) = m = =m 1 ( 1) (q m+α (v) ( 1) (q m+α (v) m (q)++1 ;q) (q;q) m (q)++1 ;q) (q;q) q (+v+1) + q (+v+1). Denote the first sum above by S 1 and the second by S. If m, then < α (v) m (q) < 1 implies that sgn(q m+α(v) m (q)++1 ; q) = ( 1) m 1. Thus sgn S 1 = ( 1) m 1, m = 1,,... In S set j = m + 1 to obtain where S = ( 1) m 1 j= ( 1) j A j, (v) A j = (qα m (q)+j ; q) q (j+m 1)(j+m+v). (q; q) j+m 1 A calculation shows that A j+1 < A j reduces to q j+m+v < (1 q α(v) m (q)+j ) (1 q m+j ), which holds because q m+v = (1 q α(v) m (q) )(1 q m ) and because the left side of the last inequality is decreasing in j and the right hand side is increasing in j. The infinite series is thus positive and therefore 4

5 sgns = ( 1) m 1 = sgns 1 From Lemma 1 and Lemma we now that J v (3) (z; q) has an odd number of roots in the interval (q m +α(v) m (q), q m ). The next theorem proves that there is exactly one root in each such interval and that there are no other roots. Theorem 3 If q v+1 < (1 q) and if w (υ) (3) (q) are the positive roots of J v (z; q),ordered increasing in, then w (v) (q) = q +ɛ(ν), with < ɛ (υ) < α (v) (q), = 1,,... Proof: From the preceding lemmas we now that v the form w (v) = q +ɛ, with < ɛ < α (v) (q). To simplify the notation we will set F (z) = (q;q) (z; q). (q ν+1 ;q) z ν v (z; q) has roots of We will prove that the only positive roots of F (z) inside the dis z < q m are the w (υ) (q), = 1,,...m. Suppose there are other roots ±λ, = 1,,..., P m ; λ >. By Jensens Theorem [1] we can write: 1 π log F (q m e iθ ) dθ π m P m = = log q m w + m log q m + ɛ + = m + m log q log q log q m λ P m log q m λ m ɛ + P m log q m λ (3) On the other hand, by the definition of the q-bessel function, we have: F (q m e iθ ) = ( 1) q (+1) m e iθ (q v+1 ; q) (q; q) = q m +m =( 1) m e imθ (q v+1 ; q) m (q; q) m 5 ( 1) q (+1) e iθ. (q m+v+1 ; q) (q m+1 ; q) = m

6 Then we have So that, lim log F (q m e iθ ) = m + m log q log(q v+1 ; q) m log(q; q) m + log ( 1) q (+1) e iθ (q m+v+1 ; q) (q m+1 ; q). 1 π π Now observe that lim = m = m log F (q m e iθ ) m + m dθ = lim + lim 1 π π log = m ( 1) q (+1) e iθ = (q m+v+1 ; q) (q m+1 ; q) log q log(q v+1 ; q) log ( 1) q (+1) e iθ (q m+v+1 ; q) (q m+1 ; q) dθ. = ( 1) q (+1) e iθ [4] The limit above is uniform in θ. By the Jacobi Triple Product Identity = ( q 1 e iθ ) (q 1 ) = (q; qe iθ ; e iθ ; q). Using the uniform convergence to interchange the limit and the integral: lim = 1 π 1 π π π log ( 1) q (+1) e iθ (q m+v+1 ; q) (q m+1 ; q) dθ = m log (q; qe iθ ; e iθ ; q) dθ π = log + 1 π 1 = log + π = log j= log (qe iθ ; q) 1 dθ + π π π log (1 q j+1 e iθ ) dθ + 6 log (e iθ ; q) dθ j= 1 π log (1 q j e iθ ) dθ π

7 The integrals in the next to the last equality above vanish because of the mean value theorem for harmonic functions. We have thus concluded that lim 1 π log F (q m e iθ ) m + m dθ = lim log q log(q v+1 ; q). π From (3) we can write: Pm lim ( log q m λ ) lim ( log q m ɛ ) = log(q v+1 ; q) (4) But, as can be seen by the Taylor expansion of α v (q), ɛ = O(q ) and this implies ɛ <. Also P m log q m λ > log q m λ 1 as m. So the identity (4) can only hold if the first sum is empty, and the only roots are thus ± w (υ) (q), = 1,,... Remar 1. It follows from (4) that = ɛ = log(qν+1 ;q) log q. Remar. Observe that for fixed j, we can always choose m sufficiently large so that q m+j+v+1 < (1 q m+j+1 )(1 q j+1 ) and q j+m+v < (1 q αv m(q)+j )(1 q m+j ). Thus w m q m+ 1 when m without the restriction q υ+1 < (1 q). In [5] Ismail conjectured that 1) lim q m w m (υ) (q) = 1 w (υ) m+1 (q ) w (υ) m (q ) ) lim = 1. q The asymptotic relation above establishes these conjectures. 7

8 w (υ) Remar 3. Lemmas 1 and and Theorem 3 state that the roots (q) satisfy the inequalities q m +α(υ) m (q) < w (υ) m (q) < q m. These bounds are quite accurate. This is evident if we estimate the length of the interval containing the roots. A somewhat tedious calculation with Taylor series shows that q m q m +α(υ) m (q) = q m +υ O(1). Clearly, for fixed q satisfying the conditions of the theorem, the bounds become increasingly accurate as either m or υ increases. References [1] L. V. Ahlfors, Complex Analysis, McGraw Hill (1979). [] J. Bustoz and J.L.Cardoso, Basic Analog of Fourier Series on a q-linear grid, Journal of Approximation Theory, 11(1), [3] J. Bustoz and S.K.Suslov, Basic Analog of Fourier Series on a q- Quadratic Grid, Methods and Applications of Analysis, 5 (1998), [4] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge (1999). [5] M. E. H. Ismail, Some Properties of Jacsons Third q-bessel Function, to appear. [6] M. E. H. Ismail, The Zeros of Basic Bessel Functions, the Functions J υ+ax (x), and Associated Orthogonal Polynomials, Journal of Mathematical Analysis and Applications, 86 (198), [7] F.H. Jacson, On Generalized Functions of Legendre and Bessel, Transactions of the Royal Society of Edinburgh, 41(194), 1-8. [8] H. T. Koelin, Some Basic Lommel Polynomials, Journal of Approximation Theory, 96(1999),

9 [9] H. T. Koelin and R. F. Swarttouw, On the zeros of the Hahn-Exton q-bessel Function and Associated q-lommel Polynomials, Journal of Mathematical Analysis and Applications, 186(1994), [1] R. Swarttouw, The Hahn-Exton q-bessel Function, thesis, Technische Universiteit Delft,