We find approximate expressions x(k,n) and y(k,n) for the real and imaginary
parts of the kth zero z_k=x_k+i y_k of the Bessel polynomial y_n(x). To obtain
these closed-form formulas we use the fact that the points of well-defined
curves in the complex plane are limit points of the zeros of the normalized
Bessel polynomials. Thus, these zeros are first computed numerically through an
implementation of the electrostatic interpretation formulas and then, a fit to
the real and imaginary parts as functions of k and n is obtained. It is shown
that the resulting complex number x(k,n)+i y(k,n) is O(1/n^2)-convergent to z_k
for fixed kComment: 9 pages, 2 figure

Calderon, Marisol L.

Campos, Rafael G.

English

arXiv

We find approximate expressionsx̃(k,n,a)andỹ(k,n,a)for the real and imaginary parts of thekth zerozk=xk+iykof the Bessel polynomialyn(x;a). To obtain these closed-form formulas we use the fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically through an implementation of the electrostatic interpretation formulas and then, a fit to the real and imaginary parts as functions ofk,nandais obtained. It is shown that the resulting complex numberx̃(k,n,a)+iỹ(k,n,a)isO(1/n2)-convergent tozkfor fixedk.</jats:p

Rafael G. Campos

Marisol L. Calderón

Crossref

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2012, Article ID 873078, 10 pagesdoi:10.1155/2012/873078Research ArticleApproximate Closed-Form Formulas forthe Zeros of the Bessel PolynomialsRafael G. Campos and Marisol L. Caldero´nFacultad de Ciencias Fı´sico-Matema´ticas, Universidad Michoacana, 58060 Morelia, MN, MexicoCorrespondence should be addressed to Rafael G. Campos, rcampos@umich.mxReceived 11 June 2012; Revised 10 September 2012; Accepted 23 September 2012Academic Editor: Stefan SamkoCopyright q 2012 R. G. Campos and M. L. Caldero´n. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.We find approximate expressions x˜k, n, a and y˜k, n, a for the real and imaginary parts of the kthzero zk  xk  iyk of the Bessel polynomial ynx;a. To obtain these closed-form formulas we usethe fact that the points of well-defined curves in the complex plane are limit points of the zeros ofthe normalized Bessel polynomials. Thus, these zeros are first computed numerically through animplementation of the electrostatic interpretation formulas and then, a fit to the real and imaginaryparts as functions of k, n and a is obtained. It is shown that the resulting complex numberx˜k, n, a  iy˜k, n, a is O1/n2-convergent to zk for fixed k.1. IntroductionThe polynomial solutions of the diﬀerential equationz2y′′z  az  2y′z − nn  a − 1yz  0, a > 0, z ∈ C, 1.1were studied systematically in 1 by the first time. They are named generalized Besselpolynomials and are given explicitly byynz;a n∑k0n!n  a − 1kn − k!k!(z2)k, 1.2as it can be shown in 2. Here, xk is the Pochhammer symbol and n  0, 1, . . .. Manyproperties as well as applications are associated to this equation; the traveling waves in theradial direction which are solutions of the wave equation in spherical coordinates can be2 International Journal of Mathematics and Mathematical Scienceswritten in terms of the polynomial solutions of 1.1. Also, this equation has application innetwork and filter design, isotropic turbulence fields, and more see the monograph 2 or3–14 and references therein for some other results. Among these, several results about theimportant problem concerning the location of its zeros have been obtained 8–11 and in12, explicit expressions for sum rules and for the homogeneous product sum symmetricfunctions of zeros of these polynomials are given. On the other hand, the electrostaticinterpretation of these zeros as the equilibrium configuration in the complex plane with alogarithmic electric potential and a dipole at the origin has been given in 13, and in 14it is shown that this equilibrium configuration is not stable. Thus, these cases show that it isdesirable to acquire new analytical knowledge about the location of the zeros of the Besselpolynomials.In this paper we give approximate explicit formulas for both the real and imaginaryparts of the kth zero zk  xk  iyk of ynz;a and show that the approximation order of thesenew formulas to the exact zeros of the Bessel polynomials is O1/n2 for fixed k.The approach followed in this paper is simple and based on three items. The first isthe electrostatic interpretation of the zeros of polynomials satisfying second-order diﬀerentialequations 15–17, the second is a simple curve fitting of numerical data, and the third is theknown fact that the points of well-defined curves in the complex plane are limit points of thezeros of the normalized Bessel polynomials 8–11. The formulas yielded by the electrostaticinterpretation of the zeros of Bessel polynomials are used to find them numerically as it hasbeen done previously with these and other sets of points 7–19. Several sets of zeros arecomputed in this way and the sets of real and imaginary values are fitted by polynomialsdepending on the index k whose coeﬃcients depend on n and a.2. Asymptotic Expressions for the ZerosLet zk  xk  iyk, k  1, 2, . . . , n, be the zeros of the Bessel polynomial ynz;a, orderedaccording to the imaginary part. Then, from 1.1 it follows that A procedure for obtainingthis kind of nonlinear equations for the zeros of a polynomial satisfying second and higherorder diﬀerential equations is given in 19.n∑k11zj − zk (azj  2)2z2j 0, 2.1where j  1, 2, . . . , n, that is, the real and imaginary parts of the zeros should satisfy theelectrostatic equationsn∑k1xj − xk(xj − xk)2 (yj − yk)2ax3j  2x2j  axjy2j − 2y2j2(x2j  y2j)2 0,n∑k1yj − yk(xj − xk)2 (yj − yk)2yj(ax2j  4xj  ay2j)2(x2j  y2j)2 0.2.2International Journal of Mathematics and Mathematical Sciences 30−1.5−1−0.5(z)a = 20Re200 300 400 500ka0Im200 300 400 5001−0.5−100.5a = 2(z)kb0−1−0.6−0.2−0.4−0.8−1.2−1.4(z)a = 400 100 200 300 400 500kRec1−0.5−100.5(z)a = 400 100 200 300 400 500kImd0−1−0.6−0.2−0.4−0.8−1.2−1.4(z)0 100 200 300 400 500ka = 100Ree1−0.5−100.5(z)0 100 200 300 400 500ka = 100ImfFigure 1: Real and imaginary parts of the zeros of the normalized Bessel polynomials yn2z/2na−2;afor a  2, 40, 100. They are plotted as functions of k for n  100, 200, 300, 400, 500, in gray-level intensity,from lower to higher, according to the value of n.This set of nonlinear equations can be solved by standard methods. We have used a Newtonmethod to solve them up to n  500 and a  100.Let ωk  μk  iνk be the kth zero of the normalized Bessel polynomials yn2z/2na−2;a, that is, μk  2n  a − 2xk/2 and νk  2n  a − 2xk/2. As it is shown in Figure 1,the piecewise linear interpolation of the real and imaginary parts of ωk can be fitted bypolynomials of the second and third degree in the index k.Thus, we propose the following expressionsμ˜k, n, a  a2n, ak2  a1n, ak  a0n, a,ν˜k, n, a  b3n, ak3  b2n, ak2  b1n, ak  b0n,2.34 International Journal of Mathematics and Mathematical Sciences0−0.8−0.6−0.4−0.20 100 200 300 400 500n∼μ(1,n,a)a0 100 200 300 400 500n−1.4−1.5−1.3−1.2−1.1∼μ(n/2,n,a)bFigure 2: Dependence of μ˜1, n, a and μ˜n/2, n, a on n for a  10, 20, 30, 40, 100. Plots are shown in gray-level intensity, from lower to higher, according to the value of a.for the approximate zero ω˜k  μ˜k, n, a  iν˜k, n, a to fit our data. To find the dependence ofthe coeﬃcients of these polynomials on n and a, we take into account the numerical behaviorof the data at the middle and end points.We begin by finding the coeﬃcients of the second-order polynomial giving the realpart by fitting the values of μ˜k, n, a at k  1 and k  n/2. In Figure 2 we show thedependence of μ˜1, n, a and μ˜n/2, n, a on n for some values of a.A fit of these data to the models −A/n  B and −A/n  B − 3/2 yieldsμ˜1, n, a  − 54a  860100n  50a  715, μ˜(n2, n, a) − 75n − 2a  40050n  11a  220. 2.4These conditions and the symmetry of μ˜k, n, awith respect to the middle point lead to thefollowing coeﬃcients:a2n, a p2n, arn, a, a1n, a p1n, arn, a, a0n, a p0n, arn, a, 2.5wherep2n, a  4(7500n2  50625n  850an − 694a2 − 2770a  96800),p1n, a  −4n  1(7500n2  50625n  850an − 694a2 − 2770a  96800),p0n, a  −100(130n2 − 993n − 7656)− 20(135n2  627n − 1580)a − 2297n  794a2,rn, a  5nn − 250n  11a  22020n  10a  143.2.6Now, to find the coeﬃcients of the third-order polynomial giving the imaginary part wefollow a similar procedure. The dependence of the real part of ν˜1, n, a and ∂ν˜k, n, a/∂k|kn/2 on n for some values of a is shown in Figure 3.International Journal of Mathematics and Mathematical Sciences 50 100 200 300 400 500−1−0.9−0.8−0.7−0.6−0.5(1,n,a)n∼νa00.050.10.150.20.25k(n/2,n,a)0 100 200 300 400 500n∼νbFigure 3: Dependence of ν˜1, n, a and ∂ν˜k, n, a/∂k|kn/2 on n for a  10, 20, 30, 40, 100. Plots are shownin gray-level intensity, from lower to higher, according to the value of a.Again, a fit of these data to the models A/n  B − 1 and A/n yieldsν˜1, n, a  −25n − a − 5025n − 1 ,∂ν˜k, n, a∂k∣∣∣∣kn/296na  25. 2.7In addition, we have thatν˜(n2, n, 2) 0, ν˜n, n, 2  −ν˜1, n, 2, 2.8therefore, the coeﬃcients are given byb3n, a q3n, asn, a, b2n, a q2n, asn, a,b1n, a q1n, asn, a, b0n, a q0n, asn, a,2.9whereq3n, a  −200(23n3 − 92n2  90n  4) 200(n3 − 5n2  8n − 6)a− 8(n2 − 2n  2)a2,q2n, a  100(69n4 − 255n3  186n2  92n  8)− 100(3n4 − 12n3  9n2  4n − 12)a  4(3n3 − 3n2  4)a2,q1n, a  50(21n4  54n3 − 522n2  748n − 176)− 50(3n4 − 9n3 − 6n2  26n − 24)a  2(3n3 − 6n  8)a2,6 International Journal of Mathematics and Mathematical Sciencesq0n, a  −25(25n4 − 217n3  618n2 − 660n  184)− 25(n4 − 2n3 − 7n2  16n − 12)a (n3  n2 − 4n  4)a2sn, a  25n − 12n − 22a  25.2.10Thus, the substitution of 2.10, 2.9, 2.6, and 2.5, respectively, in 2.3 yields the approxi-mate closed-form expressionsz˜k  x˜k, n, a  iy˜k, n, a 2μ˜k, n, a2n  a − 2  i2ν˜k, n, a2n  a − 2 , 2.11where k  1, 2, . . . , n, for the zeros of the unnormalized Bessel polynomial ynz;a. Theexpressions given in 2.11 converge to the zeros zk of these polynomials, as we will showin the following.3. ConvergenceFollowing 9, we defineWz e√11/z2z(1 √1  1/z2) , 3.1and denote by Γ the curve defined byΓ {z ∈ C : |Wz|  1, ∣∣arg z∣∣ ≥ π2}, 3.2which contains the limit points ω̂k of the zeros of the normalized Bessel polynomialyn2z/2na−2;a. Then, it has been proved in 8 that the zeroωk of yn2z/2na−2;aapproaches to order O1/n the limit value ω̂k, that is,|ωk − ω̂k|  O(1n), 3.3as n → ∞.Thus, if we show that |ω˜k − ω̂k|  O1/n, we will have proved that|ωk − ω˜k|  O(1n), 3.4and therefore, taking into account that ωk  2n  a − 2zk/2, the explicit expression 2.11approaches to order O1/n2 the zero zk of the Bessel polynomial ynz;a.International Journal of Mathematics and Mathematical Sciences 7To this purpose, we simply substitute the expression for ω˜k  μ˜k, n, a  iν˜k, n, agiven by 2.3 in 3.1 to obtain, after a lengthy calculation, that the expansion of Wω˜k interms of 1/n isWω˜k  1[2a2 − 100a3k − 2 − 5042k − 6725a  25√hk, a25a  25(cos2tk, a − sin2tk, a)]1nO(1n3/2),3.5wherehk, a  25  a2130  27a  300k2 (2a2 − 100a3k − 2 − 5042k − 67)2,tk, a 25  a130  27a  300k41675 − 100a  a2 − 1050k  150ak .3.6This implies that|Wω˜k|  1 O(1n), 3.7for fixed k and a. Thus, ω˜k approaches to order O1/n the Γ curve and 3.4 follows. Fromhere we have that|zk − z˜k|  O(1n2), 3.8as n → ∞. Numerical calculations confirm and extend this result. Figure 4 shows thebehavior of the maxima of |zk − z˜k| over k as they depend on n for the particular case of a  2.The numbers computed by 2.2 are taken as the exact zeros zk. A fit of these data gives 1/nawith a  1.7.4. Some Few TestsJust to give examples of the application of the approximate expression 2.11, we consider thefollowing cases.4.1. The Real ZeroA closed-form formula for the unique real zero αna of the Bessel polynomial ynz;a can beobtained by the substitution of k  n  1/2 in the real part of 2.11, x˜k, n. This givesα˜na  −4n3 17550 − 71a (−707a2  5900a  35000)7500nO(1n2), 4.18 International Journal of Mathematics and Mathematical Sciences100 200 300 400 5000.00020.00040.00060.00080.001max kn|z k−∼zk|Figure 4: Plot of the values of maxnk1|zk − z˜k | against n for the case of a  2.as our new result. In 2, 11 very accurate expressions for αna are given. Particularly, thefollowing formula:2αna −1.325486838n − 1.00628995a  1.349836480 O(12n  a − 2), 4.2is given in 11. Expanding 2/α˜na in powers of nwe find that2α˜na −1.33333n − 0.946667a  0.666667, 4.3indicating good relative agreement between the two results.4.2. Power SumsHere we carry out the corresponding multiplications and use some cases of Faulhaber’sformula. Then we compare our results with the exact ones.1 Sum of the Zeros. The simple sum of z˜k cf. 2.11 gives a complicated expressionfor the real part. However, expanding both the real and imaginary parts of this sum givess˜1n n∑k1z˜k  −1 (53a100− 7130 i32a  25)1nO(1n2). 4.4The exact result is s1n  −1, as can be seen from 1.2.2 Sum of the Squares of the Zeros. In this case we take the particular case of a  1. Forthis value we obtains˜2n n∑k1z˜2k 34695915nO(1n2). 4.5International Journal of Mathematics and Mathematical Sciences 9The exact sums2n 12n − 1 12nO(1n2)4.6can be found elsewhere 12.The use of the approximate formula 2.11 for obtaining sums of higher powers ofthese zeros is not expected to give satisfactory results, since the powers and the summagnifythe total error.5. Final CommentThe approximate formula for zk given above is far from being unique. There exist many otherfunctions to fit the zeros obtained through the electrostatic equations 2.2, and there are otherconditions to impose at the extreme andmiddle points of the fitting interval. For instance, theimaginary part y˜k, n can be fitted by a polynomial of degree 5, but this does not improvethe rate of convergence and, on the other hand, the calculations become more complicated.AcknowledgmentThe authors thank Consejo Nacional de Ciencia y Tecnologı´a for the financial support givento this project.References1 H. L. Krall and O. Frink, “A new class of orthogonal polynomials: the Bessel polynomials,” Trans-actions of the American Mathematical Society, vol. 65, pp. 100–115, 1949.2 E. Grosswald, Bessel Polynomials, vol. 698 of Lecture Notes in Mathematics, Springer, Berlin, Germany,1978.3 H. M. Srivastava, “Some orthogonal polynomials representing the energy spectral functions for afamily of isotropic turbulence fields,” Zeitschrift fu¨r Angewandte Mathematik und Mechanik, vol. 64, no.6, pp. 255–257, 1984.4 J. L. Lo´pez and N. M. Temme, “Large degree asymptotics of generalized Bessel polynomials,” Journalof Mathematical Analysis and Applications, vol. 377, no. 1, pp. 30–42, 2011.5 O¨. Eg˘eciog˘lu, “Bessel polynomials and the partial sums of the exponential series,” SIAM Journal onDiscrete Mathematics, vol. 24, no. 4, pp. 1753–1762, 2010.6 C. Berg and C. Vignat, “Linearization coeﬃcients of Bessel polynomials and properties of Studentt-distributions,” Constructive Approximation, vol. 27, no. 1, pp. 15–32, 2008.7 L. Pasquini, “Accurate computation of the zeros of the generalized Bessel polynomials,” NumerischeMathematik, vol. 86, no. 3, pp. 507–538, 2000.8 A. J. Carpenter, “Asymptotics for the zeros of the generalized Bessel polynomials,”Numerische Mathe-matik, vol. 62, no. 4, pp. 465–482, 1992.9 F. W. J. Olver, “The asymptotic expansion of Bessel functions of large order,” Philosophical Transactionsof the Royal Society of London Series A, vol. 247, pp. 328–368, 1954.10 H.-J. Runckel, “Zero-free parabolic regions for polynomials with complex coeﬃcients,” Proceedings ofthe American Mathematical Society, vol. 88, no. 2, pp. 299–304, 1983.11 M. G. de Bruin, E. B. Saﬀ, and R. S. Varga, “On the zeros of generalized Besselpolynomials I, II,”Indagationes Mathematicae, vol. 84, pp. 1–25, 1981.12 F. Ga´lvez and J. S. Dehesa, “Some open problems of generalised Bessel polynomials,” Journal of PhysicsA, vol. 17, no. 14, pp. 2759–2766, 1984.10 International Journal of Mathematics and Mathematical Sciences13 E. Hendriksen and H. van Rossum, “Electrostatic interpretation of zeros,” in Orthogonal Polynomialsand Their Applications, M. Alfaro, J. S. Dehesa, F. J. Marcellan, J. L. Rubio de Francia, and J. Vinuesa,Eds., vol. 1329 of Lecture Notes in Mathematics, pp. 241–250, Springer, Berlin, Germany, 1988.14 G. Valent and W. Van Assche, “The impact of Stieltjes’ work on continued fractions and orthogonalpolynomials: additional material,” Journal of Computational and Applied Mathematics, vol. 65, no. 1–3,pp. 419–447, 1995.15 G. Szego˝, Orthogonal Polynomials, Colloquium Publications, American Mathematical Society, Provi-dence, RI, USA, 1975.16 F. Marcella´n, A. Martı´nez-Finkelshtein, and P. Martinez-Gonzalez, “Electrostatic models for zeros ofpolynomials: old, new, and some open problems,” Journal of Computational and Applied Mathematics,vol. 207, pp. 258–272, 2007.17 R. G. Campos, “Perturbed zeros of classical orthogonal polynomials,” Boletin de la Sociedad MatematicaMexicana, vol. 5, no. 1, pp. 143–153, 1999.18 R. G. Campos, “Solving singular nonlinear two-point boundary value problems,” Boletin de la SociedadMatematica Mexicana, vol. 3, no. 2, pp. 279–297, 1997.19 R. G. Campos and L. A. Avila, “Some properties of orthogonal polynomials satisfying fourth orderdiﬀerential equations,” Glasgow Mathematical Journal, vol. 37, no. 1, pp. 105–113, 1995.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical Problems in EngineeringHindawi Publishing Corporationhttp://www.hindawi.comDifferential EquationsInternational Journal ofVolume 2014Applied MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical PhysicsAdvances inComplex AnalysisJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014OptimizationJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Operations ResearchAdvances inJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Function SpacesAbstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal of Mathematics and Mathematical SciencesHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AlgebraDiscrete Dynamics in Nature and SocietyHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Decision SciencesAdvances inDiscrete MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences

Approximate Closed-Form Formulas for the Zeros of the Bessel Polynomials

We find approximate expressions x̃(k,n,a) and ỹ(k,n,a) for the real and imaginary parts of the kth zero zk=xk+iyk of the Bessel polynomial yn(x;a). To obtain these closed-form formulas we use the fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically through an implementation of the electrostatic interpretation formulas and then, a fit to the real and imaginary parts as functions of k, n and a is obtained. It is shown that the resulting complex number x̃(k,n,a)+iỹ(k,n,a) is O(1/n2)-convergent to zk for fixed k

Approximate closed-form formulas for the zeros of the Bessel Polynomials

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