ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT. Gradimir V. Milovanović, Aleksandar S. Cvetković, and Zvezdan M.

Transcription

1 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome , DOI: 02298/PIM M ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT Gradimir V Milovanović, Aleksandar S Cvetković, and Zvezdan M Marjanović Abstract This is a continuation of our previous investigations on polynomials orthogonal with respect to the linear functional L : P C, where L = px dµx, dµx = x2 λ/2 expiζx dx, and P is a linear space of all algebraic polynomials Here, we prove an extension of our previous existence theorem for rational λ /2, 0], give some hypothesis on three-term recurrence coefficients, and derive some differential relations for our orthogonal polynomials, including the second order differential equation Introduction In this paper we continue our investigation on orthogonality with respect to the oscillatory weight functions studied in 7], 8], and 9] We are concerned with the following measure dµx = x 2 λ/2 expiζx dx, supported on the interval, ], where λ > /2 and where ζ R\{0} Evidently, the corresponding weight function wx = w ζ,λ x = x 2 λ/2 expiζx depends on two real parameters λ and ζ We investigate the questions connected with the existence of polynomials π n orthogonal with respect to the linear functional Lp] = px dµx, p P, where P is the space of all algebraic polynomials As it has been proved in 9] it is enough to consider only the case ζ > 0, and therefore we continue our discussion only for such values of ζ As we proved in 9] it can be easily inferred that the orthogonal polynomials π n exist in the case λ Q with λ > 0 and ζ is positive zero of the Bessel function of the first kind J λ The crucial ingredient in the proof is the result which states that non-trivial zeros of Bessel functions of the first kind are 2000 Mathematics Subject Classification: Primary 30C0; Secondary 33C47 The authors were supported in part by the Serbian Ministry of Science Project: Orthogonal Systems and Applications, grant number #44004 and the Swiss National Science Foundation SCOPES Joint Research Project No IB New Methods for Quadrature 49

2 50 MILOVANOVIĆ, CVETKOVIĆ AND MARJANOVIĆ transcendental provided their index is rational In this paper we extend the results from 9] for /2 < λ 0, λ Q, and ζ being the zero of the Bessel function J λ Also, in this paper, we investigate the possibility of computation of three-term recurrence coefficients for the given sequence of orthogonal polynomials {π n } n N0, as well as some consequences of the weight function w being semiclassical We also find the second order differential equation for the corresponding orthogonal polynomials 2 Existence of orthogonal polynomials According to, the moments 2 µ k = Lx k ] = x k x 2 λ/2 expiζx dx, k N 0, can be expressed in terms of Bessel functions J ν of the order ν defined by J ν z = + m=0 m z/2 ν+2m m!γν + m + Using the initial conditions 2λ µ 0 = AJ λ ζ, µ = ia ζ J λζ J λ ζ, 2λ2λ + µ 2 = A ζ 2 J λ ζ + 2λ + ] J λ ζ, ζ where A = 2/ζ λ π Γλ + /2, we can see that the moments 2 satisfy the following recurrence relation µ k+2 = k + 2λ + µ k+ + µ k + k iζ iζ µ k The following result is proved in 9] Theorem 2 The moments µ k can be expressed in the form 22 µ k = A iζ k P λ k ζj λ ζ + Q λ kζj λ ζ, k N 0, where A = 2/ζ λ π Γλ+/2, and Pk λ and Qλ k are polynomials in ζ, which satisfy the following four-term recurrence relation with the initial conditions and respectively y k+2 = k + 2λ + y k+ ζ 2 y k kζ 2 y k, P λ 0 ζ =, P λ ζ = 2λ, P λ 2 ζ = 2λ2λ + ζ 2 Q λ 0 ζ = 0, Q λ ζ = ζ, Q λ 2 ζ = 2λ + ζ, Some obvious properties of the polynomials P λ n and Q λ n, n N 0, are stated in the following lemma

3 ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT 5 Lemma 2 For the degree of the polynomials P λ n and Q λ n we have degp λ 2n, P λ 2n+ = 2n n N 0 and degq λ 2n, Q λ 2n = 2n n N The free terms in the polynomials P λ n and Q λ n are n 2λ n and zero, respectively The free term in the polynomial Q n ζ/ζ, n N, equals n 2λ + n The leading coefficients in the polynomials P λ 2n, P λ 2n, Q λ 2n+ and Q λ 2n are n, n 2nλ + n, n and n n2λ +, respectively Now we are ready to state and prove a stronger version of the existence theorem than the one given in 9] Theorem 22 Let λ Q, λ > /2 and let ζ 0 be a zero of the Bessel function J λ Then the sequence of monic polynomials {π n } n N0 orthogonal with respect to the linear functional exists Proof As in 9] our result is based upon completely algebraic facts Consider the sequence of Hankel determinants defined by µ 0 µ µ n µ µ 2 µ n n =, n N µ n µ n µ 2n 2 The sequence of orthogonal polynomials {π n } n N0 with respect to L exists provided n 0, n N see 4] Thus, let ζ be a nontrivial zero of the Bessel function J λ Then we can express the moments of the linear functional L in the following form µ k = A iζ k P λ k ζj λ ζ, k N 0 Obviously, from the Hankel determinant we can factor AJ λ ζ n+ /iζ nn+ and we are left with the sequence of Hankel determinants H n, n N, made from the new moment sequence µ k = P k λζ, k N 0, P0 λ ζ P λ ζ Pnζ λ P λ ζ P2 λ ζ Pn λ ζ H n = P nζ λ Pn λ ζ P2n 2ζ λ Accordingly, all H n, n N, are polynomials in ζ which have rational coefficients since λ Q If some H n = 0, then ζ is the zero of the polynomial H n with rational coefficients and ζ is zero of the Bessel function J λ According to 3, p 220], based on the results by ], 2] see also 6], non-trivial zeros of the Bessel functions J λ, λ Q, λ / N, are transcendental and cannot be the zeros of the polynomials with rational coefficients unless polynomials are identically equal to zero We have to prove that determinants H n are not identically equal to zero To prove this fact we emphasize that, according to Lemma 2, the free coefficient in the polynomial Pn λ equals n 2λ n, n N 0, so that the free coefficient in the

4 52 MILOVANOVIĆ, CVETKOVIĆ AND MARJANOVIĆ polynomial H n equals to the value of the Hankel determinant H n made for the sequence of moments µ k = 2λ k, k N 0 For λ > 0, we can easily recognize the Hankel determinants H n as being the same, up to multiplicative factor Γ2λ n, as the Hankel determinants for the generalized Laguerre weight x 2λ e x on 0, + Hence, none of the determinants H n cannot be equal according to the existence of the sequence of orthogonal polynomials with respect to the Laguerre weight function This establishes the proof for λ > 0 cf 9] Actually, we can calculate the determinant H n for λ > 0 in the form n l H n = n l! 2λ + l i, n N l=0 i=0 We easily inspect that the same result about the values of the Hankel determinants H n holds even in the case of /2 < λ 0 This due to the fact that Hankel determinants H n as being polynomials are analytic functions of λ as well as expressions representing their values Inspecting values of the determinants H n, n N, we easily conclude that H n 0, n N, for λ 0 This in turn implies that n 0, n N Case λ = 0 requires a special attention Consider now the case λ = 0 As we easily inspect we have that H n = 0, n N, for λ = 0 Consider determinants H n, we have H n = Γ2λ n Hn, n N Since H n, n N, are analytic functions of λ we can compute values of H n, n N, for λ = 0, as the limiting values of the expressions on the right hand side We have n lim λ 0 Γ2λn n l! l=0 l n l 2λ + l i = n! n l! l i 0 i=0 Thus, we have proved that if λ Q, λ > /2, and ζ is the zero of the Bessel function J λ, then the sequence of Hankel determinants l= n = AJ λ ζ n /iζ nn H n, n N, contains no zero entry Accordingly, the sequence of orthogonal polynomials with respect to the linear functional L exists Regarding Theorem 22, we suppose such parameters λ and ζ which provide the existence of orthogonal polynomials π n with respect to the linear functional The quasi inner-product p, q := Lpq], in our case has the property zp, q = p, zq, and because of that, the corresponding monic polynomials {π n } n N0 satisfy the fundamental three-term recurrence relation 23 π n+ x = x iα n π n x π n x, n N, with π 0 x =, π x = 0 The recursion coefficients α n and can be expressed in terms of Hankel determinants as cf 9] H n+ H n 24 H n+ H n 25 iα n = n+ n = n+ n iζ = = n+ n 2 n H n+ H n iζ 2, H 2 n i=0

5 ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT 53 where H n is defined before and n is the Hankel determinant n+ with the penultimate column and the last row removed The corresponding determinant H n is given by P0 λ ζ P λ ζ Pn 2ζ λ Pn λ ζ H n P λ ζ P2 λ ζ Pnζ λ Pn+ζ λ = P nζ λ Pn λ ζ P2n 3ζ λ P2nζ λ Although β 0 can be arbitrary, as usual it is convenient to take β 0 = µ 0 = AJ λ ζ In this case, however, the values of Hankel determinants cannot be found easily, but, it is clear that the recursion coefficients are rational functions in ζ Using our software package 5] we can generate coefficients even in symbolic form for some reasonable small values of n for n 2 see Table Table Recursion coefficients α n and for n 2 n α n 0 2λ ζ 2 4λ+λ ζ 2 ζ2λ ζ 2 2 ζ λ π Γ λ + 2 Jλ ζ ζ 2 2λ ζ 2 7+6λζ 6 4λ4+λ3+2λζ 4 +32λ 2 3+2λζ 2 32λ 3 2+λ 2ζζ 2 2λζ 4 λ5+2λζ 2 +4λ 2 ] 2+2λζ4 λ5+2λζ 2 +4λ 2 ] ζ 2 2λ+ζ 2 2 According to symbolic calculations we can conjecture that H n is a polynomial in ζ 2 of degree m, m H n = S m ζ 2 = γ ν λζ 2ν, where m = n 2 /4 for even n and m = n 2 /4 for odd n, so that, because of 25, ν=0 β 2k = S kk+ζ 2 S kk ζ 2 ζ 2 S k 2ζ 2 2, β 2k+ = S k+2ζ 2 S k2ζ 2 ζ 2 S kk+ ζ 2 2 In a similar way, we can conjecture that the numerator in 24 is a polynomial in ζ 2 of degree nn + /2, so that α 2k = H nh n+ H n+h n = V nn+/2 ζ 2, V k2k+ ζ 2 ζs k 2ζ 2 S kk+ ζ 2 2, α V k+2k+ ζ 2 2k+ = ζs kk+ ζ 2 S k+ 2ζ 2 Increasing n, the complexity of expressions for α n and increases quite rapidly On the other side, using the Chebyshev algorithm, similarly as in 7], an efficient numerical construction of recursion coefficients can be done According to µ k = k µ k, k N 0 see 22 we can prove:

6 54 MILOVANOVIĆ, CVETKOVIĆ AND MARJANOVIĆ Lemma 22 If the sequence of monic orthogonal polynomials {π n } n N0 then π n z = n π n z exists, 3 Differential-difference relations In this section we consider some differential relations for monic polynomials {π n } n N0 orthogonal with respect to the linear functional, ie, to the the quasi inner-product product p, q := Lpq] = px qx wx dx, supposing such parameters λ and ζ, which provide the existence of our orthogonal polynomials Our weight function wx = x 2 λ/2 e iζx satisfies the following Pearson type differential equation φw = ψw, φ = x 2, ψ = 2λ + x + iζ x 2 As before, we put dµx = wx dx Theorem 3 For every n N, we have 3 φπ n + ψπ n = where θ n+2 n = iζ, n+2 k=n θ k nπ k, θ n+ n = ζα n+ + α n n + 2λ +, 2θ n n = 2λ + iα n + iζ + + α 2 n, θ n n = n, and α n and are coefficients in the recurrence relation 23 Proof Since φwπ n x k dx = φwπ n x k k φπ n x k dµ = k φπ n x k dµ = 0 for k < n, in that case we conclude that φwπ n /w is orthogonal to x k Therefore, we have directly 32 φwπ n = φπ n + ψπ n = w n+2 k=n θ k nπ k, according to the fact that degψ = 2 Since we are concerned with the monic polynomials it is readily seen that θn n+2 = iζ

7 ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT 55 we get For θn n, after multiplying 32 with π n and integrating with respect to dµ, θ n n π n 2 = and accordingly θn n = n We also have θ k n π k 2 + θ n k π n 2 = φwπ n π n dx = φwπ n π ndx = n π n 2, = The last equality for k = n + gives θn n+ = θn n+ + + π n+ 2 = n + 2λ + φwπ n π k + φwπ k π n ] dx φπ n π k + φπ kπ n + ψπ k π n ]dµ = iζ π n+ 2 ψπ n π k dµ 2λ + x + iζ x 2 ] π n π n+ dµ = n + 2λ + + α n+ + α n ζ, because of xπ n = π n+ + iα n π n + π n and x 2 π n π n+ dµ = Similarly, for k = n, we get because of 2θ n n = π n 2 = π n 2 x 2 π n π n+ dµ π n+ + iα n π n + π n π n+2 + iα n+ π n+ + + π n dµ = iα n+ π n+ 2 + iα n π n 2 ψπ 2 ndµ { 2λ + = i 2λ + α n + ζ + + α 2 n], x 2 π 2 ndµ = Thus, the equality 3 is proved } xπndµ 2 + iζ π n 2 x 2 πndµ 2 π n+ + iα n π n + π n 2 dµ = π n+ 2 α 2 n π n 2 + β 2 n π n 2 Lemma 3 The monic polynomials orthogonal with respect to the linear functional satisfy the following differential-difference equation φπ n + ψπ n = p n 2 π n + q n π n, n N, where p n 2 and q n are polynomials of the second and first degree, respectively

8 56 MILOVANOVIĆ, CVETKOVIĆ AND MARJANOVIĆ Proof Starting with equation 3 and using the three-term recurrence relation 23 we get φπ n + ψπ n = θn n+2 x iα n+ + θn n+ x iα n + θn n+2 + θn] n πn + θn n+2 x iα n+ + θn n+ + θn n What is left to do is to read the respective polynomials ] πn Theorem 32 The monic polynomials orthogonal with respect to the linear functional satisfy the following equation 33 φπ n = p n π n + q n π n, where p n = nx iζ βn+ + + α 2 iα n 2 n + 2n + 2λ +, 2 q n = 2n + λ + iζx + iα n ] Proof Using expressions given in the proof of Lemma 3 and Theorem 3 we get exactly what is stated Theorem 33 The polynomials p n and q n, which appear in 33, satisfy the following recurrence relations p n+ = q n x iα n q n+ = x iα n respectively x iα n, + p n + q n p n + q n x iα n p n q n ] x iα n + φ + q n, Proof We can prove this result using almost the same arguments from the proof in 7] Namely, we have φπ n+ φπ n = x iα n φπ n φπ n = x iα n p n π n + q n π n p n π n + q n π n 2 = q n x iα n p n q n x iα n π n+ { + x iα n p n + q n } + x iα n q n x iα n p n q n x iα n π n, from where we just need to read corresponding polynomials In the sequel we derive a system of nonlinear recurrence relations for the recursion coefficients α n and, which needs a few first coefficients as starting values see Table

9 ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT 57 Theorem 34 We have 34 p n+ + p n + q n x iα n = 2λ x iζ x 2, n N Proof Starting with the previous theorem we can prove that quantity p n+ + p n + q n x iα n, is independent of n Hence, using Table, we can prove that and hence finish the proof p 2 + p + q x iα = 2λ x iζ x 2, β Theorem 35 The three-term recurrence coefficients satisfy the following system of nonlinear recurrence equations +2 = + α 2 n+ α 2 n ζ αn+ 2n + 2λ + 3 α n 2n + 2λ ], α n+2 = 2+2 2αn + + α n α 2 n+ α 2 n ] + ζ+2 α 2 n+ n + λ + /2 αn+ α n n + λ / n + λ n + λ ] Proof The first equation we prove quite easily Namely, simply compare coefficients with x 0 in 34 and we have the stated equation For the proof of the second equation we have to use the second relation given in Theorem 33 In the same way, comparing coefficients with the term x 0 in this relation we obtain the result Now, we can prove a result concerning zeros of orthogonal polynomials π n Theorem 36 The polynomials π 2n+, n N 0, have only simple zeros The even polynomials π 2n, n N, have only simple zeros except possibly one zero of the multiplicity two which is purely imaginary Proof It is a trivial fact that polynomials π n and π n cannot have zeros in common If x is multiple zero of π n, using equation 33, it must be q n x = 0 However, q n is a polynomial of first degree and, therefore, the multiplicity of the zero can be at most two Using a symmetry property of polynomials, we see that if x is a zero of the polynomial π n, then also x is the zero of this polynomial π n Hence, we conclude that π n cannot have a multiple zero outside of the imaginary axis Finally, according to the symmetry we conclude that if polynomial of odd degree has zero on the imaginary axis it must be of odd multiplicity Hence, all polynomials of odd degree have only simple zeros We can now also state a result connected with a differential equation which solutions are polynomials orthogonal with respect to the functional

10 58 MILOVANOVIĆ, CVETKOVIĆ AND MARJANOVIĆ Theorem 37 The polynomials π n orthogonal with respect to the functional satisfy the following second order differential equation φq n π n + ψq n iζ φπ n nq n iζ p n qn ˆp n q n p n ˆq n ] π n = 0, φ where we used the notation from Theorem 32 and ˆp n = q n / and ˆq n = p n 2 + x iα n q n / The term ˆp n q n p n ˆq n must have the factor φ Proof Starting with the equation 33 and the three-term recurrence relation, we compute easily φπ n = qn π n + p n + q n x iα n π n = ˆp n π n + ˆq n π n Combining this equation with the one for the polynomial π n, ie with equation 33, we get φπ φ n p n ] π n = ˆp n π n + ˆq n φπ n p n π n q n After some calculations we obtain the following differential equation φq n π n + q n φ p n ˆq n φq n ] π n + p n q n + p n q n q n ˆp n q n p n ˆq n φ q n ] π n = 0 If we use Theorems 34 and 32, for the term with π n we got exactly what is stated Similarly we have the same for the term with π n Finally, it is clear that, since all other quantities are polynomials it must be that q n ˆp n q n p n ˆq n /φ is polynomial, too Obviously it cannot be that q n has zeros in common with φ since it can have only purely imaginary zero This theorem can be used to give the following result: Theorem 38 Let n be an odd integer and let x n ν, ν =,, n, be simple zeros of the polynomial π n Then 35 π nx n ν π nx n ν + λ + /2 x n ν + λ + /2 x n ν + + iζ x n ν + iα n + 2n + 2λ/iζ = 0, for each ν =,, n Proof We start with the differential equation for polynomials π n, derived in the previous theorem We divide it with π nx n ν 0, ν =,, n This equality is fulfilled due to the fact that for odd n, according to Theorem 36, the polynomial π n has only simple zeros Thus, by division and using the fact that π n x n ν = 0, ν =,, n, we have π n π n + ψ φ iζβ n q n x n ν = 0, ν =,, n, which after small calculation becomes exactly what is stated

11 ORTHOGONAL POLYNOMIALS FOR THE OSCILLATORY-GEGENBAUER WEIGHT 59 This theorem enables the computation of zeros of orthogonal polynomial π n of the high degree using technique similar as given in 0] 4 Zero distribution Here we want to present some consequences of Theorem 38 in the light of some new results of Aptekarev and Van Assche see ] Namely, provided that the sequence of polynomials π n orthogonal with respect to the functional exists, they have proved that for λ = 0 the three term recurrence coefficients satisfy the following asymptotic properties α n 0 and /4 Actually, we can use this fact to give some interesting observations connected to the zero distribution of polynomials π n A direct consequence of the mentioned theorem is the boundness of the zeros x n ν, ν =,, n, of polynomials π n uniformly in n N see 2], 3] Figure Zero distribution of π n, n = 0000, for λ = and ζ a zero of J Figure 2 Zero distribution of π n, n = 0000, for λ = and ζ a zero of J 0 According to the mentioned result and for n sufficiently large, our equation 35 becomes π n x n ν + λ + 2 x n ν + x n + iζ + On 0, ν + π n

12 60 MILOVANOVIĆ, CVETKOVIĆ AND MARJANOVIĆ for each ν =,, n If we compare it to the similar equation for the classical Gegenbauer weight w 0,λ see 4], we notice that there is only the difference in the term iζ + On Taking the imaginary part we get 2 ν k Imx k x ν x k x ν 2 + λ + 2 Imx k x n ν 2 + Imx k x n k + 2 ζ + On 0, which describes a property of zeros Namely, for a fixed n and bigger ζ, zeros are at the bigger distance from the x axis as presented in Figures and 2 References ] A I Aptekarev, W V Assche, Scalar and matrix Riemann-Hilbert approach to the strong asymptotics of Pade approximants and complex orthogonal polynomials with varying weight, J Appox Theory , ] B Beckermann, Complex Jacobi matrices, J Comput Appl Math , ] B Beckerman, M Castro, On the determinacy of the comoplex Jacobi operators, Publication ANNO 446, Universite Lille, ] T S Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 978 5] A S Cvetković, G V Milovanović, The Mathematica package OrthogonalPolynomials, Facta Univ Ser Math Inform , ] N I Fel dman, A B Šidlovskiĭ, The development and present state of the theory of transcendental numbers, Uspehi Mat Nauk 22:3 967, Russian Engl transl in Russian Math Surveys 22:3 967, 79] 7] G V Milovanović, A S Cvetković, Orthogonal polynomials and Gaussian quadrature rules related to oscillatory weight functions, J Comput Appl Math , ] G V Milovanović, A S Cvetković, Orthogonal polynomials related to the oscillatory- Chebyshev weight function, Bull Cl Sci Math Nat Sci Math , ] G V Milovanović, A S Cvetković, M P Stanić, Orthogonal polynomial and quadrature rules realted to Oscillatory-Gegenabuer weight function submitted; preprint version available at: 0] L Pasquini, Accurate computation of the zeros of the generalized Bessel polynomials, Numer Math , ] C L Siegel, Über einige Anwendungen diophantischer Approximationen, Abh Preuss Akad Wiss, Phys-Math 929, 70 Gesammelte Abhandlungen Springer-Verlag, Berlin New York 966 Band I, ] 2] C L Siegel, Transcendental Numbers, Annals of Mathematics Studies 6, Princeton University Press, Princeton, N J, 949 3] A B Šidlovskiĭ, Transcendental Numbers, Nauka, Moscow, 987 in Russian 4] G Szegő, Orthogonal Polynomials, Amer Math Soc Colloq Publ 23, 4th ed, Amer Math Soc, Providence, R I, 975 Faculty of Computer Sciences, Megatrend University Beograd, Serbia gvm@megatrend-edunet Faculty of Science and Mathematics, University of Niš Niš, Serbia aca@elfakniacyu Faculty of Electronic Engineering, University of Niš Niš, Serbia zvezdan@elfakniacyu