Panayiotis D. Siafarikas: His Life and Work

Transcription

1 Advances in Dynamical Systems and Applications ISSN , Volume 8, Number 2, pp (2013) Panayiotis D. Siafarikas: His Life and Work Evangelos K. Ifantis and Chrysi G. Kokologiannaki University of Patras Department of Mathematics Patras, Greece Eugenia N. Petropoulou University of Patras Department of Engineering Sciences Division of Applied Mathematics & Mechanics Patras, Greece Abstract In this paper, the life and work of the late Professor Panayiotis D. Siafarikas is presented. His research interests lied within differential equations, difference equations, special functions and orthogonal polynomials. As a minimum tribute, a conference dedicated to his memory was organized in Patras, Greece, in September 2012, entitled International Conference on Differential Equations, Difference Equations and Special Functions. This paper summarizes the welcome talks presented at the conference which covered several aspects of his personal, academic and research life. AMS Subject Classifications: 01A70. Keywords: Functional-analytic methods, Bessel functions, orthogonal polynomials, difference equations, differential equations, complex domain. 1 P. D. Siafarikas: His Personal and Professional Life Panayiotis (Panos) D. Siafarikas was born on 2 February 1953, at the village Tirnavos in Thessaly, Central Greece, but his origins were from Avdela, a small village up in the Received November 15, 2012; Accepted November 30, 2012 Communicated by Ondřej Došlý

2 170 E. K. Ifantis, C. G. Kokologiannaki and E. N. Petropoulou Northern Pindus mountain range in Western Macedonia (Greece), which is situated at an altitude of 1350 meters. He finished high school in Athens and he received his diploma from the Department of Mathematics of the University of Patras in 1976 and his Ph. D. from the same department in 1980, under the supervision of Professor Evangelos K. Ifantis. He was elected lecturer at the Department of Mathematics of the University of Patras on 1984, passed through all academic positions and became full professor at the same department on He was married to Katerina Rekka, with whom he had three children: Athina, Dimitris Panayiotis and Aimilia. He died in July Panayiotis efficiently served his department from various academic and administrative positions: He was elected Chairman of the Division of Applied Analysis for 5 years, Deputy Chairman of the department for 6 years and Chairman of the department for 4 years. He has also served the Hellenic Open University (H. O. U.) for 5 years, from 2004 until 2009, as the President of its Board of Directors. It was during his presidency, that the H. O. U. was relocated in its new privately owned buildings. He was also an active member of the scientific society and visiting professor in various universities abroad. He has participated in many national and international conferences, not only by presenting his research results, but also as a member of the corresponding scientific committee or as an invited speaker. He was also the main organizer of three conferences held in Patras: the 4th National Conference on Mathematical Analysis (1994), the 5th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA, 1999) and the International Conference on Differential, Difference Equations and Applications (in honor of Prof. E. K. Ifantis, 2002). His research work covered several areas of mathematics such as differential equations (mostly ordinary), ordinary and partial difference equations, Bessel functions, orthogonal polynomials and mathematical physics. He was the author or coauthor of 72 research papers published in international refereed scientific journals and 6 university books dedicated to ordinary differential equations and their applications [41, 43], abstract analysis [1], integral equations [42], Bessel functions and orthogonal polynomials [40], as well as ordinary difference equations and their applications [29]. He was also the guest editor of the proceedings of two international conferences [34, 35] and member of the editorial board of 12 international scientific journals: Nonlinear Dynamics and System Theory, Archives of Inequalities and Applications, Journal of Inequalities and Applications, Advances in Differences Equations, Journal of Applied Functional Analysis, Journal of the Applied Mathematics, Statistics & Informatics, International Journal of Difference Equations, Bulletin of the Greek Mathematical Society, Romanian Society of Applied and Industrial Mathematics, International Journal of Modern Mathematics, International Journal of Differential Equations, Advances and Applications in Mathematical Sciences. His work was appreciated by the rest of the academic community as well as by the rest of the society and this is depicted not only by the references that his work has and is still attracting, but also by the prizes and honors he had received. More precisely,

3 Panayiotis D. Siafarikas: His Life and Work 171 he has received: the first prize in the VI Balkaniade of Young Researchers in 1982, the F. Pallas prize (jointly with Eugenia N. Petropoulou) from the Academy of Athens, in 2003 and the Premio della Simpatia from the municipality of Rome in He also became Doctor Honoris Causa of the University of SS Cyril and Methodius in Trnava, in Last, but not least, it should be mentioned that Professor Siafarikas was an excellent teacher, always willing to help the young students and researchers. His friends, students and collaborators, will always remember him with love and affection and will be grateful for everything that he has taught and offered them. 2 The Research Work of Professor Siafarikas Panayiotis started, as a PhD student of Professor Ifantis, from differential equations. More precisely, he started from ordinary singular differential equations in the complex domain. At the time, an interesting problem was the problem of finding conditions under which an ordinary singular differential equation or singular system of differential equations can have analytic solutions [2, 47]. The most known results were local, i.e., the most of the known conditions predicted analytic solutions in a neighborhood of zero. Some mathematicians asked for solutions in the class of functions which are analytic in the open unit disc and continuously differentiable on the boundary [3, 4]. Panayiotis [36] asked for solutions in the Hardy Lebesgue space H 2 ( ), i.e., the Hilbert space of functions f(z) = a n z n 1 which are analytic in the open unit disc = {z C : z < 1} and satisfy the condition or equivalently the condition sup 0<r<1 a n 2 < n=1 2π 0 f(re iθ ) dθ <. Panayiotis followed a functional analytic method introduced in [5], which reduces the problem of finding solutions of differential equations or systems of differential equations to an operator problem in an abstract Hilbert space. Often, the operator under consideration is a Fredholm operator and the theory of Fredholm operators can successfully be applied. Panos research on this field at the beginning of his career is published in [36 39]. A singular differential equation of particular interest during that period, which was also studied in [3], was: n=1 z 2 y (z) + (a 0 + a 1 z)y(z) = h(z), (2.1)

4 172 E. K. Ifantis, C. G. Kokologiannaki and E. N. Petropoulou where a 0, a 1 are complex numbers and h(z) = h n z n 1 is an analytic function in some neighborhood of zero. Panos et. al. studied (2.1) in [16] and they proved, among other things, the following theorem, which gives necessary and sufficient conditions so that (2.1) to have a solution in H 2 ( ): Theorem 2.1 (See [16]). Suppose h(z) H 2 ( ). Then, equation (2.1) has a solution in H 2 ( ) if and only if: (a) h(0) = 0 and dk+1 h(z) = h (k+1) (0) = 0, in the case where a 0 = 0 and z=0 dz k+1 a 1 = k, k = 0, 1, 2,.... (b) h(0) = 0, in the case where a 0 = 0 and a 1 k, k = 0, 1, 2,.... n=1 (c) n=1 ( 1) n 1 α n 1 0 h n Γ(α 1 + n 1) = 0, in the case where a 0 0. The necessary and sufficient condition (c) of Theorem 2.1, was also found in [3, 4, 47] for specific values of a 0 and a 1 and it concerned solutions of (2.1), which are analytic in and continuously differentiable on the boundary of, provided that h(z) is also analytic in and continuously differentiable on the boundary of. By choosing a 0 = ρ/2, a 1 = µ + 1 and h(z) = ρ ( 2 exp ρz ) = ρ 2 2 n=1 ( 1) n 1 (n 1)! ( ρz ) n 1, 2 it follows (due to Theorem 2.1(c)) that equation z 2 y (z) + ( ρ ) 2 + (µ + 1)z y(z) = ρ ( 2 exp ρ ) 2 z, (2.2) has a solution in H 2 ( ) if and only if ( ρ ) 2n+µ ( 1) n 1 2 Γ(n + 1)Γ(n + µ + 1) = 0 J µ(ρ) = 0, n=0 i.e., if and only if ρ is a zero of the Bessel function J µ (z), where ρ and µ are in general complex numbers! Furthermore, the authors of [16], transformed (2.2) into an equivalent operator equation in an abstract separable Hilbert space H with orthonormal base {e n }, n = 1, 2,... and connected the problem of the existence of solutions in H 2 ( ) of (2.2), not only with the zeros of the Bessel function J µ (z), but also with the eigenvalues of specific operators. More precisely, it was proved that:

5 Panayiotis D. Siafarikas: His Life and Work 173 Theorem 2.2 (See [16]). The complex number ρ 0 is a zero of the Bessel function J µ (z), with µ C, if and only if the number 2/ρ is an eigenvalue of the operator where L µ is the diagonal operator A µ = L µ T 0, L µ e n = 1 n + µ e n, n = 1, 2,... and T 0 = V + V, where V is the shift operator and V its adjoint defined by: V e n = e n+1, n = 1, 2, 3,... V e n = e n 1, n = 2, 3,..., V e 1 = 0. Notice that, although n µ so that L µ to be well defined, this is not a restriction, because due to the well-known relation J µ (z) = ( 1) µ J µ (z) (see e.g., [48, p. 15]), the Bessel functions J µ (z) and J µ (z) have the same zeros. Especially in the case where µ is a real number with µ > 1, the square root L 1/2 µ of L µ exists and it is defined by Then, the following holds: L 1/2 µ e n = 1 n + µ e n, n = 1, 2,.... Theorem 2.3 (See [16]). The number ρ 0 is a zero of the Bessel function J µ (z), with order µ > 1, if and only if the number 2/ρ is an eigenvalue of the operator S µ = L 1/2 µ A µ L 1/2 µ. This operator S µ is a compact, self-adjoint and Hilbert Schmidt operator. Due to these properties of S µ, several already known properties of the Bessel functions were obtained in [16], such as Lommel Hurwitz theorem (see e.g., [48, p. 478]) and Rayleigh s formula (see e.g., [48, p. 502]). Many other properties for the zeros of Bessel functions were easily obtained and some of them were published not as new results, but as an application of the necessary and sufficient condition of Theorem 2.1(c), which has been found to be new and hard to be proved. However, Panos and Professor Ifantis insisted and after a short time, a new differential equation which the roots of Bessel functions satisfy, has been discovered in [11]. This differential equation was the subject of several investigations and generalizations [9, 12, 18]. Many bounds which are new or improve well-known bounds (see [12]), followed from the differential equation which the roots of Bessel functions satisfy. Two of these bounds were successfully used to prove a result for the imaginary roots of the second

6 174 E. K. Ifantis, C. G. Kokologiannaki and E. N. Petropoulou derivative of Bessel functions. It was known, that the second derivative of J ν (x) possesses the imaginary roots ±iρ(ν) and it was found by numerical calculations, that one of them, in a small interval of the positive real axis increases, reaches its maximum and then decreases [21]. This has been proved analytically in [14]. Several important results concerning the complex zeros of Bessel functions and other related functions were obtained in [22 24]. Panos occupation with the zeros of Bessel functions and related topics lasted many years and resulted in 21 research papers. His last paper in the area of Bessel functions was [30], where the common zeros of J ν (z) for fixed z or fixed ν were investigated. In this investigation, Theorem 2.2 was used, together with results from semigroup theory. Parallel to his PhD work, Panos at the beginning of his career, has been occupied also with subjects of mathematical physics which led to 7 research papers. Indicatively, his first [19] and last paper [20] on this field, are mentioned. Another theme with which Panayiotis has been occupied for many years, was the subject of orthogonal polynomials. His research on this field resulted in 22 papers, most of which concern bounds and monotonicity properties for their zeros, as well as information about the measure or measures of orthogonality. As it is well-known (see e.g., [17] or [46]), orthogonal polynomials P n (x) satisfy a three term recurrence relation of the form α n P n+1 (x) + β n P n (x) + a n 1 P n 1 (x) = xp n (x), n = 1, 2,..., N P 0 (x) = 0, P 1 (x) = 1, (2.3) where α n > 0 and β n are real sequences. Together with Professor Ifantis, Panos used a functional-analytic method for the study of the zeros of orthogonal polynomials. It is well-known that the zeros of P N+1 (x) satisfying (2.3), are the same as the eigenvalues of a symmetric, tridiagonal, Jacobi matrix. In [13], a different decomposition of this Jacobi matrix was used, namely the tridiagonal operator T = AV + V A + B, was used, where A, B are the diagonal operators: Ae k = α k e k, Be k = β k e k, k = 1,..., N and V, V are the truncated shift operators: V e k = e k+1, k = 1,..., N 1, V e n = 0, V e k = e k 1, k = 2,..., N, V e 1 = 0, where {e k } N k=1 is an orthonormal base of an abstract, finite dimensional Hilbert space H N. Moreover, if the sequences α n, β n depend on a parameter, say λ, then the operators A and B, depend also on this parameter λ. In this way, the q parameter of q-orthogonal polynomials or the c parameter of associated polynomials, can be incorporated in the diagonal operators A and B and thus, these classes of polynomials can be studied essentially with the same functional-analytic method. This has been done for example

7 Panayiotis D. Siafarikas: His Life and Work 175 in [10,15,25,31,45]. It is worth mentioning, that this method can also be used for other polynomials satisfying more general recurrence relations than (2.3). Although Panos was occupied (roughly speaking from 1982 until 2000) with research in the field of special functions and orthogonal polynomials, he did not forget differential equations. He was kept constantly informed on the research regarding analytic solutions of differential equations and, even published a couple of papers regarding analytic and entire solutions of linear ordinary differential equations during that period. Panayiotis came back to differential equations and studied also difference equations at the end of the previous century. He studied nonlinear difference and differential equations, by reducing initial value problems of these equations to abstract operator problems and then using fixed point theorems to establish existence and uniqueness of solutions. The most important is that he, in collaboration with his student Eugenia Petropoulou, extended the method developed by Ifantis for linear and nonlinear ordinary difference [8] and differential equations [6, 7] to partial difference [26, 27] and differential equations. Maybe the most important of these results, is the proof of a simple necessary and sufficient condition, so that a quite general class of linear partial differential equations to have polynomial solutions [28]. Another important achievement, especially from the point of view of applications, was obtained by Panayiotis in collaboration with Eugenia Petropoulou and Efstratios Tzirtzilakis in [32, 33]. These papers concern a new discretization technique of ordinary differential equations. By use of this technique, the ordinary differential equation under consideration is transformed into an equivalent difference equation but not an approximate one, as it is usual the case with the standard discretization schemes. This equivalent difference equation is then used in order to calculate the solution of the differential equation. This technique is a functional-analytic one and it is actually a combination of the methods already used by Panayiotis and his collaborators for the study of differential and difference equations. This new discretization was successfully applied to initial value problems and boundary value problems (in combination with a standard shooting technique) of the Duffing equation, the Blasius equation and the Lorenz system. It has also been applied for the investigation of the behavior of the corresponding solutions in the complex plane. The obtained numerical results were compared with the 4th order Runge Kutta method and it is indicated that this new technique is better than the Runge Kutta method, with respect to the accuracy and the CPU time required. Moreover, this new technique is independent of the grid used. Panayiotis last paper, Analytic and periodic solutions for systems of differential equations, was written during the period he was ill, but it was published after his death in 2011 in the Journal of Applied Mathematics, Statistics and Informatics [44]. Research was still giving him comfort until the very end. If we would like to summarize Panos attitude towards research, maybe there is no better way than using his favorite phrase: In research, you ve got to have the two p s: patience and persistence.

8 176 E. K. Ifantis, C. G. Kokologiannaki and E. N. Petropoulou References [1] C. Drossos and P. D. Siafarikas. Basic abstract analysis. Patras, Greece, 2nd edition, (In Greek). [2] L. J. Grimm and L. M. Hall. Holomorphic solutions of functional differential systems near singular points. Proc. Amer. Math. Soc., 42: , [3] L. J. Grimm and L. M. Hall. An alternative theorem for singular differential systems. J. Differential Equations, 18(2): , [4] L. M. Hall. A characterization of the cokernel of a singular Fredholm differential operator. J. Differential Equations, 24(1):1 7, [5] E. K. Ifantis. An existence theory for functional-differential equations and functional-differential systems. J. Differential Equations, 29(1):86 104, [6] E. K. Ifantis. Analytic solutions for nonlinear differential equations. J. Math. Anal. Appl., 124(2): , [7] E. K. Ifantis. Global analytic solutions of the radial nonlinear wave equation. J. Math. Anal. Appl., 124(2): , [8] E. K. Ifantis. On the convergence of power series whose coefficients satisfy a Poincare-type linear and nonlinear difference equation. Complex Variables Theory Appl., 9(1):63 80, [9] E. K. Ifantis. A theorem concerning differentiability of eigenvectors and eigenvalues with some applications. Appl. Anal., 28(4): , [10] E. K. Ifantis, C. G. Kokologiannaki, and P. D. Siafarikas. Newton sum rules and monotonicity properties of the zeros of scaled co-recursive associated polynomials. Methods Appl. Anal., 3(4): , [11] E. K. Ifantis and P. D. Siafarikas. A differential equation for the zeros of Bessel functions. Applicable Anal., 20(3 4): , [12] E. K. Ifantis and P. D. Siafarikas. Ordering relations between the zeros of miscellaneous Bessel functions. Appl. Anal., 23(1 2):85 110, [13] E. K. Ifantis and P. D. Siafarikas. Differential inequalities for the largest zero of Laguerre and Ultraspherical polynomials. Actas del VI Symposium on Polinomios Orthogonales y Applicationes, Gijon, Spain, [14] E. K. Ifantis and P. D. Siafarikas. A result on the imaginary zeros of J ν (z). J. Approx. Theory, 62(2): , 1990.

9 Panayiotis D. Siafarikas: His Life and Work 177 [15] E. K. Ifantis and P. D. Siafarikas. Differential inequalities and monotonicity properties of the zeros of associated Laguerre and Hermite polynomials. Ann. Numer. Math., 2(1 4):79 91, [16] E. K. Ifantis, P. D. Siafarikas, and C. B. Kouris. Conditions for solution of a linear first-order differential equation in the Hardy Lebesgue space and applications. J. Math. Anal. Appl., 104(2): , [17] M. E. H. Ismail. Classical and quantum orthogonal polynomials in one variable. With two chapters by Walter Van Assche. With a foreword by Richard A. Askey, volume 98 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, [18] M. E. H. Ismail and M. E. Muldoon. On the variation with respect to a parameter of zeros of Bessel and q-bessel functions. J. Math. Anal. Appl., 135(1): , [19] A. D. Jannussis, L. C. Papaloucas, and P. D. Siafarikas. Eigenfunctions and eigenvalues of the q-differential operators. Hadronic J., 3(6): , 1979/80. [20] A. D. Jannussis, P. D. Siafarikas, P. Fillipakis, T. Fillipakis, V. Papatheou, A. Leodaris, V. Zisis, and N. Tsangas. Damped and coupled oscillators. Hadronic J., 7: , [21] M. K. Kerimov and S. L. Skorokhodov. Calculation of multiple zeros of derivatives of cylindrical Bessel functions J ν (z) and Y ν (z). Zh. Vychisl. Mat. i Mat. Fiz., 25(12): , (In Russian). [22] C. G. Kokologiannaki, M. E. Muldoon, and P. D. Siafarikas. A unimodal property of purely imaginary zeros of Bessel and related functions. Canad. Math. Bull., 37(3): , [23] C. G. Kokologiannaki and P. D. Siafarikas. Nonexistence of complex and purely imaginary zeros of a transcendental equation involving Bessel functions. Z. Anal. Anwendungen, 10(4): , [24] C. G. Kokologiannaki, P. D. Siafarikas, and C. B. Kouris. On the complex zeros of H µ (z), J µ(z), J µ(z) for real or complex order. J. Comput. Appl. Math., 40(3): , [25] C. G. Kokologiannaki, P. D. Siafarikas, and J. D. Stabolas. Monotonicity properties and inequalities of the zeros of q-associated polynomials. Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday, 1, 2: , Kluwer Acad. Publ., Dordrecht.

10 178 E. K. Ifantis, C. G. Kokologiannaki and E. N. Petropoulou [26] E. N. Petropoulou and P. D. Siafarikas. Bounded solutions of a class of linear delay and advanced partial difference equations. Dynam. Systems Appl., 10(2): , [27] E. N. Petropoulou and P. D. Siafarikas. Solutions of nonlinear delay and advanced partial difference equations in the space l 1. Comput. Math. Appl., 45(6 9): , [28] E. N. Petropoulou and P. D. Siafarikas. Polynomial solutions of linear partial differential equations. Commun. Pure Appl. Anal., 8(3): , [29] E. N. Petropoulou and P. D. Siafarikas. Difference equations and applications. Patras, Greece, 2nd edition, (In Greek). [30] E. N. Petropoulou, P. D. Siafarikas, and I. D. Stabolas. On the common zeros of Bessel functions. J. Comput. Appl. Math., 153(1 2): , [31] E. N. Petropoulou, P. D. Siafarikas, and I. D. Stabolas. Convexity results for the largest zero and functions involving the largest zero of q-associated polynomials. Integral Transforms Spec. Funct., 16(2): , [32] E. N. Petropoulou, P. D. Siafarikas, and E. E. Tzirtzilakis. A discretization technique for the solution of ODEs. J. Math. Anal. Appl., 331(1): , [33] E. N. Petropoulou, P. D. Siafarikas, and E. E. Tzirtzilakis. A discretization technique for the solution of ODEs II. Numer. Funct. Anal. Optim., 30(5 6): , [34] (Edt.) P. D. Siafarikas. Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications. Dedicated to Professor Theodore Chihara. Held in Patras, Greece, September 20 24, 1999, volume 133(1 2) of J. Comput. Appl. Math. Elsevier Science, [35] (Edt.) P. D. Siafarikas. Proceedings of the International Conference on Differential, Difference Equations and Applications. In honor of Professor Evangelos Ifantis. Held in Patras, Greece, July 1 5, 2002, volume 2004 of Abstr. Appl. Anal. Hindawi Publishing Corporation, [36] P. D. Siafarikas. Analytic solutions of singular linear differential equations and applications. PhD thesis, Department of Mathematics, University of Patras, Patras, Greece, (In Greek). [37] P. D. Siafarikas. A singular functional-differential equation. Internat. J. Math. Math. Sci., 5(3): , 1982.

11 Panayiotis D. Siafarikas: His Life and Work 179 [38] P. D. Siafarikas. Conditions for analytic solutions of a singular differential. Applicable Anal., 17(1):1 12, [39] P. D. Siafarikas. On the number of analytic solutions of a singular differential system. Complex Variables Theory Appl., 4(1):49 56, [40] P. D. Siafarikas. Special functions. Patras University Press, Patras, Greece, (In Greek). [41] P. D. Siafarikas. Applications of ordinary differential equations. Volume I. Patras, Greece, 3rd edition, (In Greek). [42] P. D. Siafarikas. Integral equations. Patras University Press, Patras, Greece, 2nd edition, (In Greek). [43] P. D. Siafarikas. Applications of ordinary differential equations. Volume II. Patras, Greece, 2nd edition, (In Greek). [44] P. D. Siafarikas. Analytic and periodic solutions for systems of differential equations. J. Appl. Math. Stat. Inform. (JAMSI), 7(1):5 23, [45] P. D. Siafarikas, I. D. Stabolas, and L. Velázquez. Differential inequalities of functions involving the lowest zero of some associated orthogonal q-polynomials. Integral Transforms Spec. Funct., 16(4): , [46] G. Szegö. Orthogonal polynomials, volume XXIII of Colloquium Publications. American Mathematical Society, Providence, R.I., 4th edition, [47] H. L. Turrittin. My mathematical expectations. Symposium on Ordinary Differential Equations (Univ. Minnesota, Minneapolis, Minn., 1972; dedicated to Hugh L. Turrittin), volume 312 of Lecture Notes in Math., pages Springer, Berlin, [48] G. N. Watson. A treatise on the theory of Bessel functions. Reprint of the second (1944) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995.