SOME RELATIONS ON HERMITE MATRIX POLYNOMIALS. Levent Kargin and Veli Kurt
|
|
- Allan George
- 5 years ago
- Views:
Transcription
1 Mathematical ad Computatioal Applicatios, Vol. 18, No. 3, pp , 013 SOME RELATIONS ON HERMITE MATRIX POLYNOMIALS Levet Kargi ad Veli Kurt Departmet of Mathematics, Faculty Sciece, Uiversity of Adeiz TR-07058, Atalya, Turey. Abstract- I this study we give additio theorem, multiplicatio theorem ad summatio formula for Hermite matrix polyomials. We write Hermite matrix polyomials as hypergeometric matrix fuctios. We also obtai a ew geeratig fuctio for Hermite matrix polyomials ad usig this fuctio, we prove some ew results ad relatios. Key Words- Hypergeometric matrix fuctios, Hermite matrix polyomials, geeratig matrix fuctios. 1. INTRODUCTION I the recet two decades, orthogoal matrix polyomials comprise a emergig field of study, with importat results i both theory ad applicatios cotiuig to appear i the literature. Hermite matrix polyomials are itroduced by Jodar ad Compay i [9]. Moreover, some properties of the Hermite matrix polyomials are give i [3, 14, 15] ad a geeralized form of the Hermite matrix polyomials have bee itroduced ad studied i [16, 17, 19, 0,, 4]. Other classical orthogoal polyomials as Laguerre, Gegebauer, Chebyshev ad Jacobi polyomials have bee exteded to orthogoal matrix polyomials, ad some results have bee ivestigated, see for example [4, 5, 8, 1, 3]. From the coectio with orthogoal matrix polyomials, special matrix fuctios have bee itroduced ad studied by some mathematicias. Gamma matrix fuctio is itroduced ad studied i [7, 10] for matrices i C r r whose eigevalues are all i the right ope half-plae. Apart from the close relatioships with the well-ow beta ad gamma matrix fuctios, the emergig theory of orthogoal matrix polyomials ad its operatioal calculus suggest the study of hypergeometric matrix fuctio. Hypergeometric matrix fuctio F ; A; z has bee recetly itroduced i [13]. Explicit closed form for geeral solutios of the hypergeometric matrix differetial equatios is give i [1]. The paper is orgaized as follows. I the ext sectio we deal with importat properties of the Hermite matrix polyomials such as additio, multiplicatio theorems ad summatio formula. We obtai a geeratig fuctio for Hermite matrix polyomials ad write these polyomials as hypergeometric matrix fuctios. We obtai some results which follow from this geeratig fuctio. Throughout this paper, if A is a matrix i C r r, its spectrum A will deotes the set of all the eigevalues of A. Its -orm will be deoted by A ad defied by A sup Ax, x 0 x where for a y i C r, y y T, y 1 is the Euclidea orm of y. If f z ad g z are holomorphic fuctios of the complex variable z, which are defied i a ope set Ω of
2 34 L. Kargi ad V. Kurt the complex plae, ad A is a matrix i C r r such that σ A Ω, the from the properties of matrix fuctioal calculus [6, page 558], it follows that f A g A g A f A. If D 0 is the complex plae cut alog the egative real axis ad log z deotes the priciple logarithm of z, the z 1 represets exp 1 log z. If A is a matrix Cr r i which σ A D 0 the A 1 A deotes the image by z 1 of the matrix fuctioal calculus actig o the matrix A. Let A be a matrix i C r r such that Re z > 0 for every eigevalues z σ A, (1) The the Hermite matrix polyomials H x, A are defied by [9] H x, A 1! x A!!, 0 () ad satisfyig the three-terms recurrece relatio H x, A x AH 1 x, A 1 H x, A, 1. H 1 x, A 0, H 1 x, A I, where I is the uit matrix i C r r. Accordig to [9], we have 0 H x, A! t exp xt A t. (3) The Pochhammer symbol or shifted factorial is defied by [11] A A A + I A + ( 1 I), 1, (4) with A 0 I. By usig (4) it is easy to show that A A A + I. (5) The hypergeometric matrix fuctio F A, B; C; z has bee give i [11] A B C 1 F A, B; C; z z,! z < 1, 0 where A, B, C are matrices i C r r such that C + I is ivertible for all itegers 0. Note that by (4) if A ii where i is a atural umber the A i+j 0 for j 1 ad F A, B; C; z becomes a matrix polyomial of degree i. Lemma 1: ([18]) Let. deotes ay matrix orm for which I 1. If M < 1 for a matrix M i C r r, the I + M c exists ad give by I M c c M,! where c is a positive iteger. We coclude this sectio recallig a result related to the rearragemet of the terms i iterated series. If A, ad B, are matrices i C r r for 0, 0, the i a aalogous way to the proof of Lemma 11 of [], it follows that 0
3 Some Relatios o Hermite Matrix Polyomials 35 ad 0 0 A, B, 0 0 A,, B,. (6) (7). SOME RELATIONS ON HERMITE MATRIX POLYNOMIALS Propositio : Hermite matrix polyomials satisfy the multiplicatio ad additio formula as follows: H x, A!!! + H z 1 + z + 1, A where ad are costats. 1 1 H x, A, (8) H z 1, A H z, A (9) Proof: Taig x for x ad t for t i (3), we have 0 H x, A t! exp xt A t exp xt A t + t t H x, A 1 1 t +!!. 0 By usig (6) ad comparig the coefficiets of t o both sides of the above equatio,! we get (8). We ca similarly prove the equatio (9). Corollary 3: The Hermite matrix polyomials have the followig relatios: H z 1 + z, A H x, A H x + y, A H z 1, A H z, A, (10) H x, A H y, A, (11) H x, A H y, A. (1) Propositio 4: The summatio formulas for H x, A are give as follows:
4 36 L. Kargi ad V. Kurt ad for x y, m 0 H x, A H m x, A m!! AH m x, A H m (y, A) m +1 m! mi m, H m+ x, A m!!! H +1 y, A H x, A H +1 x, A H y, A +1! (y x) (13). (14) Proof: Usig (3), we have 0 m 0 s H x, A H m x, A u! v m! exp ( Ax u + v (u + v) + uv m u m v m ı m! 0 m0 H ı x, A (u + v).! Maig ecessary arragemet ad comparig the coefficiets of u completes the! m! proof of (13). Equatio (14) ca be easily proved by usig the three-terms recurrece relatio for Hermite matrix polyomials. Propositio 5: Let A be a matrix i C r r satisfyig coditio (1) ad v m A < 1. The Hermite matrix polyomials have the followig geeratig fuctio: (c) H (x, A) t (I xt A) c F ci!, c + 1 I ; ; 4t I xt A, 0 where c is a positive iteger ad t < 1, x < 1. Proof: Usig () ad (6), we have 0 (c) H (x, A) t! By usig Lemma 1, we have (c) H (x, A) t! ( 1) (c)! ( 1) (c) (x A)!! t ( 1) (c + I) (xt A) (c) t.!! (I xt A) c+ t. Usig the relatio (5) ad maig ecessary arragemet, completes the proof. Theorem 6: Let A be a ivertible matrix i C r r satisfyig coditio (1). The Hermite matrix polyomials H x, A ca be write as hypergeometric matrix fuctios: H x, A x A F I, 1 I ; ; A 1 x. (15) Proof: From (), we have
5 Some Relatios o Hermite Matrix Polyomials 37 H x, A x A ( 1) ( I) (A)! x.! Sice ( I)!, usig the relatio (5) we get (15). Theorem 7: Let A be a ivertible matrix i C r r satisfyig coditio (1). For Z +, Hermite matrix polyomials have the followig geeratig fuctio: 0 H + x, A t! exp(xt A t )H x A Proof : By usig (7), we have 0 H + x, A t! u! 0 0 H x, A exp(xt A t ) H x, A t u!! t + u! 1 H x A u By comparig the coefficiets of, we obtai (16).! As a example of equatio (16), let us derive the followig theorem: t, A. (16) 1 t, A u!. Theorem 8: Let A be a ivertible matrix i C r r satisfyig coditio (1). Hermite matrix polyomials satisfy the followig relatio: t 1 H x, A H y, A! 1 4t exp A (xyt (x + y )t ) ı ı 1 4t, (17) where t < 1. Proof: By usig (6) ad (16), we have 0 Sice H x, A H y, A t! ( 1) x A H (y, A)t! ( )! H + y, A x A! ı ( 1) t! exp Axyt Ax t H y xt, A! ı ı ( 1) t.
6 38 L. Kargi ad V. Kurt H y xt, A ad! 1! 1 0 H x, A H y, A t! s0 ( 1) s! y xt s A, it follows that s! s! (1 4t ) 1 exp Axyt Ax t exp Combiig the expoetial factors, we arrive at (17). s At (y xt) ı 1 4t. Theorem 9: Let A be a matrix i C r r satisfyig coditio (1), A < 1 ad c be a positive iteger. Hermite matrix polyomials satisfy the followig relatio: 0 F I, c; ; y H x, A t! exp (xt A t )(I + xyt A yt ) c F ci, c + 1 I ; ; 4y t (I + xyt A yt ). Proof: Applyig equatio (16) to Propositio 5, we complete the proof. 3. CONCLUSIONS I this paper, we carry the properties of classical scalar Hermite polyomials to the Hermite matrix polyomials. Equatio (1) is the matrix aalog of the Ruge additio formula of the scalar Hermite polyomials. For the case A 1 1, the expressio (13) cocides with the formula which was proved by Feldheim for classical scalar Hermite polyomials. Also Propositio 5 is the matrix aalogous of the Batema s geeratig relatio for classical scalar Hermite polyomials give i [1]. Replacig t with t i (17), we give aother proof for the equatio (41) i [14]. Theorem 9 is the matrix aalog of the Brafma s relatio for classical scalar Hermite polyomials i []. Acowledgemets- The preset ivestigatio was supported by the Scietific Research Project Admiistratio of Adeiz Uiversity. 4. REFERENCES 1. F. Brafma, Geeratig fuctios of Jacobi ad related polyomials, Proceedigs of the America Mathematical Society (6), , F. Brafma, Some geeratig fuctios for Laguerre ad Hermite polyomials, Caadia Joural of Mathematics 9, , E. Defez ad L. Jodar, Some applicatios of the Hermite matrix polyomials series expasios, Joural of Computatioal Applied Mathematics 99, , E. Defez ad L. Jodar, Chebyshev matrix polyomials ad secod order matrix differetial equatios, Utilitas Mathematica 6, , 00.,
7 Some Relatios o Hermite Matrix Polyomials E. Defez ad L. Jodar, Jacobi Matrix Differetial Equatio, Polyomial Solutios, ad their Properties, Computers ad Mathematics with Applicatios 48, , N. Duford ad J. Schwartz, Liear Operators, Vol. I, Itersciece, New Yor, L. Jodar, R. Compay ad E. Posoda, Orthogoal matrix polyomials ad systems of secod order differetial equatios, Differetial Equatios ad Dyamical Systems 3 (3), 69-88, L. Jodar, R. Compay ad E. Navarro: Laguerre matrix polyomials ad systems of secod order differetial equatios, Applied Numerical Mathematics 15, 53-63, L. Jodar ad R. Compay: Hermite matrix polyomials ad secod order matrix differetial equatios, Approximatio Theory ad its Applicatios 1 (), 0-30, L. Jodar, J. C. Cortes, Some properties of gamma ad beta fuctios, Applied Mathematics Letters 11(1), 89-93, L. Jodar ad J. C. Cortes, O the hypergeometric matrix fuctio, Joural of Computatioal ad Applied Mathematics 99, 05-17, L. Jodar ad J. C. Cortes, Closed form solutio of the hypergeometric matrix differetial equatio, Mathematical ad Computer Modellig 3, , L. Jodar ad J. Sastre, O the Laguerre matrix polyomials, Utilitas Mathematica 53, 37-48, L. Jodar, E. Defez, Some ew matrix formulas related to hermite matrix polyomials theory, Proceedigs of the Iteratial Worshop o Orthogoal Polyomials i Mathematical Physics, Legaes, L. Jodar ad E. Defez, O Hermite matrix polyomials ad Hermite matrix fuctio, Approximatio Theory ad its Applicatios 14 (1), 36-48, P. Lacaster, Theory of Matrices, Academic Press, New Yor, M. S. Metwally, M. T. Mohamed ad A. Shehata, O Hermite-Hermite matrix polyomials, Mathematica Bohemica 133 (4), , M. S. Metwally, M. T. Mohamed ad A. Shehata, Geeralizatios of two-idex twovariable, Hermite matrix polyomials, Demostratio Mathematica 4(4), , M. S. Metwally, M. T. Mohamed ad A. Shehata, O pseudo Hermite matrix polyomials of two variables, Baach Joural of Mathematical Aalysis 4(), , E. D. Raiville, Special Fuctios, Chelsea, New Yor, K. A. M. Sayyed, M. S. Metwally ad R.S. Bataha, Gegebauer matrix polyomials ad secod order matrix differetial equatios, Divulgacioes Matematicas 1() , K. A. M. Sayyed, M. S. Metwally ad R. S. Bataha, O geeralized Hermite matrix polyomials, Electroic Joural of Liear Algebra 10, 7-79, 003.
Generating Functions for Laguerre Type Polynomials. Group Theoretic method
It. Joural of Math. Aalysis, Vol. 4, 2010, o. 48, 257-266 Geeratig Fuctios for Laguerre Type Polyomials α of Two Variables L ( xy, ) by Usig Group Theoretic method Ajay K. Shula* ad Sriata K. Meher** *Departmet
More informationReview Article Incomplete Bivariate Fibonacci and Lucas p-polynomials
Discrete Dyamics i Nature ad Society Volume 2012, Article ID 840345, 11 pages doi:10.1155/2012/840345 Review Article Icomplete Bivariate Fiboacci ad Lucas p-polyomials Dursu Tasci, 1 Mirac Ceti Firegiz,
More informationResearch Article Approximate Riesz Algebra-Valued Derivations
Abstract ad Applied Aalysis Volume 2012, Article ID 240258, 5 pages doi:10.1155/2012/240258 Research Article Approximate Riesz Algebra-Valued Derivatios Faruk Polat Departmet of Mathematics, Faculty of
More informationTHE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES
Joural of Mathematical Aalysis ISSN: 17-341, URL: http://iliriascom/ma Volume 7 Issue 4(16, Pages 13-19 THE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES ABEDALLAH RABABAH, AYMAN AL
More informationShivley s Polynomials of Two Variables
It. Joural of Math. Aalysis, Vol. 6, 01, o. 36, 1757-176 Shivley s Polyomials of Two Variables R. K. Jaa, I. A. Salehbhai ad A. K. Shukla Departmet of Mathematics Sardar Vallabhbhai Natioal Istitute of
More informationA solid Foundation for q-appell Polynomials
Advaces i Dyamical Systems ad Applicatios ISSN 0973-5321, Volume 10, Number 1, pp. 27 35 2015) http://campus.mst.edu/adsa A solid Foudatio for -Appell Polyomials Thomas Erst Uppsala Uiversity Departmet
More information1 6 = 1 6 = + Factorials and Euler s Gamma function
Royal Holloway Uiversity of Lodo Departmet of Physics Factorials ad Euler s Gamma fuctio Itroductio The is a self-cotaied part of the course dealig, essetially, with the factorial fuctio ad its geeralizatio
More informationTaylor polynomial solution of difference equation with constant coefficients via time scales calculus
TMSCI 3, o 3, 129-135 (2015) 129 ew Treds i Mathematical Scieces http://wwwtmscicom Taylor polyomial solutio of differece equatio with costat coefficiets via time scales calculus Veysel Fuat Hatipoglu
More informationA Note On The Exponential Of A Matrix Whose Elements Are All 1
Applied Mathematics E-Notes, 8(208), 92-99 c ISSN 607-250 Available free at mirror sites of http://wwwmaththuedutw/ ame/ A Note O The Expoetial Of A Matrix Whose Elemets Are All Reza Farhadia Received
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationA note on the p-adic gamma function and q-changhee polynomials
Available olie at wwwisr-publicatioscom/jmcs J Math Computer Sci, 18 (2018, 11 17 Research Article Joural Homepage: wwwtjmcscom - wwwisr-publicatioscom/jmcs A ote o the p-adic gamma fuctio ad q-chaghee
More informationNumerical Solution of the Two Point Boundary Value Problems By Using Wavelet Bases of Hermite Cubic Spline Wavelets
Australia Joural of Basic ad Applied Scieces, 5(): 98-5, ISSN 99-878 Numerical Solutio of the Two Poit Boudary Value Problems By Usig Wavelet Bases of Hermite Cubic Splie Wavelets Mehdi Yousefi, Hesam-Aldie
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationOn Bilateral Generating Relation for a Sequence of Functions
It. Joural of Math. Aalysis, Vol. 2, 2008, o. 8, 383-389 O Bilateral Geeratig Relatio for a Sequece of Fuctios A. K. Shukla ad J. C. Prajapati Departmet of Mathematics S.V. Natioal Istitute of Techology,
More informationOn Tricomi and Hermite-Tricomi Matrix Functions of Complex Variable
Communications in Mathematics and Applications Volume (0), Numbers -3, pp. 97 09 RGN Publications http://www.rgnpublications.com On Tricomi and Hermite-Tricomi Matrix Functions of Complex Variable A. Shehata
More informationAPPROXIMATION BY BERNSTEIN-CHLODOWSKY POLYNOMIALS
Hacettepe Joural of Mathematics ad Statistics Volume 32 (2003), 1 5 APPROXIMATION BY BERNSTEIN-CHLODOWSKY POLYNOMIALS E. İbili Received 27/06/2002 : Accepted 17/03/2003 Abstract The weighted approximatio
More informationBenaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco
EXTENDING THE BERNOULLI-EULER METHOD FOR FINDING ZEROS OF HOLOMORPHIC FUNCTIONS Beaissa Beroussi Uiversité Abdelmalek Essaadi, ENSAT de Tager, B.P. 416, Tager, Morocco e-mail: Beaissa@fstt.ac.ma Mustapha
More informationVienna, Austria α n (1 x 2 ) n (x)
ON TURÁN S INEQUALITY FOR LEGENDRE POLYNOMIALS HORST ALZER a, STEFAN GERHOLD b, MANUEL KAUERS c2, ALEXANDRU LUPAŞ d a Morsbacher Str. 0, 5545 Waldbröl, Germay alzerhorst@freeet.de b Christia Doppler Laboratory
More informationGENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION
J Korea Math Soc 44 (2007), No 2, pp 487 498 GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION Gi-Sag Cheo ad Moawwad E A El-Miawy Reprited from the Joural of the Korea Mathematical
More informationA 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
More informationResearch Article Some E-J Generalized Hausdorff Matrices Not of Type M
Abstract ad Applied Aalysis Volume 2011, Article ID 527360, 5 pages doi:10.1155/2011/527360 Research Article Some E-J Geeralized Hausdorff Matrices Not of Type M T. Selmaogullari, 1 E. Savaş, 2 ad B. E.
More informationREGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS
REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x]
More informationON GENERALIZATION OF SISTER CELINE S POLYNOMIALS
Palestie Joural of Mathematics Vol. 5() (6), 5 Palestie Polytechic Uiversity-PPU 6 ON GENERALIZATION OF SISTER CELINE S POLYNOMIALS Khursheed Ahmad, M. Kamarujjama ad M. Ghayasuddi Commuicated by Jose
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationThe Riemann Zeta Function
Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we
More informationInfinite Series and Improper Integrals
8 Special Fuctios Ifiite Series ad Improper Itegrals Ifiite series are importat i almost all areas of mathematics ad egieerig I additio to umerous other uses, they are used to defie certai fuctios ad to
More informationGAMALIEL CERDA-MORALES 1. Blanco Viel 596, Valparaíso, Chile. s: /
THE GELIN-CESÀRO IDENTITY IN SOME THIRD-ORDER JACOBSTHAL SEQUENCES arxiv:1810.08863v1 [math.co] 20 Oct 2018 GAMALIEL CERDA-MORALES 1 1 Istituto de Matemáticas Potificia Uiversidad Católica de Valparaíso
More informationSPECTRUM OF THE DIRECT SUM OF OPERATORS
Electroic Joural of Differetial Equatios, Vol. 202 (202), No. 20, pp. 8. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.ut.edu ftp ejde.math.txstate.edu SPECTRUM OF THE DIRECT SUM
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationEXPANSION FORMULAS FOR APOSTOL TYPE Q-APPELL POLYNOMIALS, AND THEIR SPECIAL CASES
LE MATEMATICHE Vol. LXXIII 208 Fasc. I, pp. 3 24 doi: 0.448/208.73.. EXPANSION FORMULAS FOR APOSTOL TYPE Q-APPELL POLYNOMIALS, AND THEIR SPECIAL CASES THOMAS ERNST We preset idetities of various kids for
More informationModified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations
Noliear Aalysis ad Differetial Equatios, Vol. 5, 27, o. 4, 57-7 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ade.27.62 Modified Decompositio Method by Adomia ad Rach for Solvig Noliear Volterra Itegro-
More information-ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION
NEW NEWTON-TYPE METHOD WITH k -ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION R. Thukral Padé Research Cetre, 39 Deaswood Hill, Leeds West Yorkshire, LS7 JS, ENGLAND ABSTRACT The objective
More informationIntroduction to Optimization Techniques
Itroductio to Optimizatio Techiques Basic Cocepts of Aalysis - Real Aalysis, Fuctioal Aalysis 1 Basic Cocepts of Aalysis Liear Vector Spaces Defiitio: A vector space X is a set of elemets called vectors
More informationConcavity Solutions of Second-Order Differential Equations
Proceedigs of the Paista Academy of Scieces 5 (3): 4 45 (4) Copyright Paista Academy of Scieces ISSN: 377-969 (prit), 36-448 (olie) Paista Academy of Scieces Research Article Cocavity Solutios of Secod-Order
More informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
More informationMatrix representations of Fibonacci-like sequences
NTMSCI 6, No. 4, 03-0 08 03 New Treds i Mathematical Scieces http://dx.doi.org/0.085/tmsci.09.33 Matrix represetatios of Fiboacci-like sequeces Yasemi Tasyurdu Departmet of Mathematics, Faculty of Sciece
More informationSome properties of Boubaker polynomials and applications
Some properties of Boubaker polyomials ad applicatios Gradimir V. Milovaović ad Duša Joksimović Citatio: AIP Cof. Proc. 179, 1050 (2012); doi: 10.1063/1.756326 View olie: http://dx.doi.org/10.1063/1.756326
More informationEVALUATION OF SUMS INVOLVING PRODUCTS OF GAUSSIAN q-binomial COEFFICIENTS WITH APPLICATIONS
EALATION OF SMS INOLING PRODCTS OF GASSIAN -BINOMIAL COEFFICIENTS WITH APPLICATIONS EMRAH KILIÇ* AND HELMT PRODINGER** Abstract Sums of products of two Gaussia -biomial coefficiets are ivestigated oe of
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationNumerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION
alaysia Joural of athematical Scieces 3(1): 83-93 (9) umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod 1 Ali Hassa ohamed urid ad Teh Yua Yig 1, Departmet of athematics, Faculty
More informationUniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations
Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume 3 Issue Versio 0 Year 03 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic (USA Olie
More informationNew Generalization of Eulerian Polynomials and their Applications
J. Aa. Num. Theor. 2, No. 2, 59-63 2014 59 Joural of Aalysis & Number Theory A Iteratioal Joural http://dx.doi.org/10.12785/jat/020206 New Geeralizatio of Euleria Polyomials ad their Applicatios Sera Araci
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationAnalytical solutions for multi-wave transfer matrices in layered structures
Joural of Physics: Coferece Series PAPER OPEN ACCESS Aalytical solutios for multi-wave trasfer matrices i layered structures To cite this article: Yu N Belyayev 018 J Phys: Cof Ser 109 01008 View the article
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationThe r-generalized Fibonacci Numbers and Polynomial Coefficients
It. J. Cotemp. Math. Scieces, Vol. 3, 2008, o. 24, 1157-1163 The r-geeralized Fiboacci Numbers ad Polyomial Coefficiets Matthias Schork Camillo-Sitte-Weg 25 60488 Frakfurt, Germay mschork@member.ams.org,
More informationWHAT ARE THE BERNOULLI NUMBERS? 1. Introduction
WHAT ARE THE BERNOULLI NUMBERS? C. D. BUENGER Abstract. For the "What is?" semiar today we will be ivestigatig the Beroulli umbers. This surprisig sequece of umbers has may applicatios icludig summig powers
More informationTRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES. 1. Introduction
Math Appl 6 2017, 143 150 DOI: 1013164/ma201709 TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES PANKAJ KUMAR DAS ad LALIT K VASHISHT Abstract We preset some iequality/equality for traces of Hadamard
More informationGeneralized Little q-jacobi Polynomials as Eigensolutions of Higher-Order q-difference Operators
Geeralized Little q-jacobi Polyomials as Eigesolutios of Higher-Order q-differece Operators Luc Viet Alexei Zhedaov CRM-2583 December 1998 Cetre de recherches mathématiques, Uiversité de Motréal, C.P.
More informationAdvanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology
Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4
More informationk-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
More informationSOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS
Folia Mathematica Vol. 5, No., pp. 4 6 Acta Uiversitatis Lodziesis c 008 for Uiversity of Lódź Press SOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS ROMAN WITU LA, DAMIAN S
More information(p, q)-type BETA FUNCTIONS OF SECOND KIND
Adv. Oper. Theory 6, o., 34 46 http://doi.org/.34/aot.69. ISSN: 538-5X electroic http://aot-math.org p, q-type BETA FUNCTIONS OF SECOND KIND ALI ARAL ad VIJAY GUPTA Commuicated by A. Kamisa Abstract. I
More informationAbstract. 1. Introduction This note is a supplement to part I ([4]). Let. F x (1.1) x n (1.2) Then the moments L x are the Catalan numbers
Abstract Some elemetary observatios o Narayaa polyomials ad related topics II: -Narayaa polyomials Joha Cigler Faultät für Mathemati Uiversität Wie ohacigler@uivieacat We show that Catala umbers cetral
More informationOn Generalized Fibonacci Numbers
Applied Mathematical Scieces, Vol. 9, 215, o. 73, 3611-3622 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.5299 O Geeralized Fiboacci Numbers Jerico B. Bacai ad Julius Fergy T. Rabago Departmet
More informationNumerical integration of analytic functions
Numerical itegratio of aalytic fuctios Gradimir V. Milovaović, Dobrilo Ð Tošić, ad Miloljub Albijaić Citatio: AIP Cof. Proc. 1479, 146 212); doi: 1.163/1.4756325 View olie: http://dx.doi.org/1.163/1.4756325
More informationFLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS. H. W. Gould Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS H. W. Gould Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV
More informationON THE HADAMARD PRODUCT OF BALANCING Q n B AND BALANCING Q n
TWMS J App Eg Math V5, N, 015, pp 01-07 ON THE HADAMARD PRODUCT OF ALANCING Q AND ALANCING Q MATRIX MATRIX PRASANTA KUMAR RAY 1, SUJATA SWAIN, Abstract I this paper, the matrix Q Q which is the Hadamard
More informationBangi 43600, Selangor Darul Ehsan, Malaysia (Received 12 February 2010, accepted 21 April 2010)
O Cesáro Meas of Order μ for Outer Fuctios ISSN 1749-3889 (prit), 1749-3897 (olie) Iteratioal Joural of Noliear Sciece Vol9(2010) No4,pp455-460 Maslia Darus 1, Rabha W Ibrahim 2 1,2 School of Mathematical
More informationA GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca
Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity
More informationA Characterization of Compact Operators by Orthogonality
Australia Joural of Basic ad Applied Scieces, 5(6): 253-257, 211 ISSN 1991-8178 A Characterizatio of Compact Operators by Orthogoality Abdorreza Paahi, Mohamad Reza Farmai ad Azam Noorafa Zaai Departmet
More informationEigenvalue localization for complex matrices
Electroic Joural of Liear Algebra Volume 7 Article 1070 014 Eigevalue localizatio for complex matrices Ibrahim Halil Gumus Adıyama Uiversity, igumus@adiyama.edu.tr Omar Hirzallah Hashemite Uiversity, o.hirzal@hu.edu.jo
More informationDirect Estimates for Lupaş-Durrmeyer Operators
Filomat 3:1 16, 191 199 DOI 1.98/FIL161191A Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: http://www.pmf.i.ac.rs/filomat Direct Estimates for Lupaş-Durrmeyer Operators
More informationAn almost sure invariance principle for trimmed sums of random vectors
Proc. Idia Acad. Sci. Math. Sci. Vol. 20, No. 5, November 200, pp. 6 68. Idia Academy of Scieces A almost sure ivariace priciple for trimmed sums of radom vectors KE-ANG FU School of Statistics ad Mathematics,
More informationSeries with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers
3 47 6 3 Joural of Iteger Sequeces, Vol. 5 (0), Article..7 Series with Cetral Biomial Coefficiets, Catala Numbers, ad Harmoic Numbers Khristo N. Boyadzhiev Departmet of Mathematics ad Statistics Ohio Norther
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationWeighted Approximation by Videnskii and Lupas Operators
Weighted Approximatio by Videsii ad Lupas Operators Aif Barbaros Dime İstabul Uiversity Departmet of Egieerig Sciece April 5, 013 Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig
More informationThe Arakawa-Kaneko Zeta Function
The Arakawa-Kaeko Zeta Fuctio Marc-Atoie Coppo ad Berard Cadelpergher Nice Sophia Atipolis Uiversity Laboratoire Jea Alexadre Dieudoé Parc Valrose F-0608 Nice Cedex 2 FRANCE Marc-Atoie.COPPO@uice.fr Berard.CANDELPERGHER@uice.fr
More informationA q 2 -Analogue Operator for q 2 -Analogue Fourier Analysis
JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS, 5758 997 ARTICLE NO AY975547 A -Aalogue Operator for -Aalogue Fourier Aalysis Richard L Rubi Departmet of Mathematics, Florida Iteratioal Uiersity, Miami,
More informationAN ALMOST LINEAR RECURRENCE. Donald E. Knuth Calif. Institute of Technology, Pasadena, Calif.
AN ALMOST LINEAR RECURRENCE Doald E. Kuth Calif. Istitute of Techology, Pasadea, Calif. form A geeral liear recurrece with costat coefficiets has the U 0 = a l* U l = a 2 " ' " U r - l = a r ; u = b, u,
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationThe Binet formula, sums and representations of generalized Fibonacci p-numbers
Europea Joural of Combiatorics 9 (008) 70 7 wwwelseviercom/locate/ec The Biet formula, sums ad represetatios of geeralized Fiboacci p-umbers Emrah Kilic TOBB ETU Uiversity of Ecoomics ad Techology, Mathematics
More informationAccepted in Fibonacci Quarterly (2007) Archived in SEQUENCE BALANCING AND COBALANCING NUMBERS
Accepted i Fiboacci Quarterly (007) Archived i http://dspace.itrl.ac.i/dspace SEQUENCE BALANCING AND COBALANCING NUMBERS G. K. Pada Departmet of Mathematics Natioal Istitute of Techology Rourela 769 008
More informationBESSEL- AND GRÜSS-TYPE INEQUALITIES IN INNER PRODUCT MODULES
Proceedigs of the Ediburgh Mathematical Society 007 50, 3 36 c DOI:0.07/S00309505000 Prited i the Uited Kigdom BESSEL- AND GRÜSS-TYPE INEQUALITIES IN INNER PRODUCT MODULES SENKA BANIĆ, DIJANA ILIŠEVIĆ
More informationPoincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains
Advaces i Pure Mathematics 23 3 72-77 http://dxdoiorg/4236/apm233a24 Published Olie Jauary 23 (http://wwwscirporg/oural/apm) Poicaré Problem for Noliear Elliptic Equatios of Secod Order i Ubouded Domais
More informationImproving the Localization of Eigenvalues for Complex Matrices
Applied Mathematical Scieces, Vol. 5, 011, o. 8, 1857-1864 Improvig the Localizatio of Eigevalues for Complex Matrices P. Sargolzaei 1, R. Rakhshaipur Departmet of Mathematics, Uiversity of Sista ad Baluchesta
More informationEnumerative & Asymptotic Combinatorics
C50 Eumerative & Asymptotic Combiatorics Notes 4 Sprig 2003 Much of the eumerative combiatorics of sets ad fuctios ca be geeralised i a maer which, at first sight, seems a bit umotivated I this chapter,
More informationCOMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q n } Sang Pyo Jun
Korea J. Math. 23 2015) No. 3 pp. 371 377 http://dx.doi.org/10.11568/kjm.2015.23.3.371 COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q } Sag Pyo Ju Abstract. I this ote we cosider a geeralized
More informationA Mean Paradox. John Konvalina, Jack Heidel, and Jim Rogers. Department of Mathematics University of Nebraska at Omaha Omaha, NE USA
A Mea Paradox by Joh Kovalia, Jac Heidel, ad Jim Rogers Departmet of Mathematics Uiversity of Nebrasa at Omaha Omaha, NE 6882-243 USA Phoe: (42) 554-2836 Fax: (42) 554-2975 E-mail: joho@uomaha.edu A Mea
More informationON SOME TRIGONOMETRIC POWER SUMS
IJMMS 0: 2002 185 191 PII. S016117120200771 http://ijmms.hidawi.com Hidawi Publishig Corp. ON SOME TRIGONOMETRIC POWER SUMS HONGWEI CHEN Received 17 Jue 2001 Usig the geeratig fuctio method, the closed
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationSolution of Differential Equation from the Transform Technique
Iteratioal Joural of Computatioal Sciece ad Mathematics ISSN 0974-3189 Volume 3, Number 1 (2011), pp 121-125 Iteratioal Research Publicatio House http://wwwirphousecom Solutio of Differetial Equatio from
More informationEnumerative & Asymptotic Combinatorics
C50 Eumerative & Asymptotic Combiatorics Stirlig ad Lagrage Sprig 2003 This sectio of the otes cotais proofs of Stirlig s formula ad the Lagrage Iversio Formula. Stirlig s formula Theorem 1 (Stirlig s
More information#A51 INTEGERS 14 (2014) MULTI-POLY-BERNOULLI-STAR NUMBERS AND FINITE MULTIPLE ZETA-STAR VALUES
#A5 INTEGERS 4 (24) MULTI-POLY-BERNOULLI-STAR NUMBERS AND FINITE MULTIPLE ZETA-STAR VALUES Kohtaro Imatomi Graduate School of Mathematics, Kyushu Uiversity, Nishi-ku, Fukuoka, Japa k-imatomi@math.kyushu-u.ac.p
More informationHarmonic Number Identities Via Euler s Transform
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810
More informationCommon Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex
More informationDominant of Functions Satisfying a Differential Subordination and Applications
Domiat of Fuctios Satisfyig a Differetial Subordiatio ad Applicatios R Chadrashekar a, Rosiha M Ali b ad K G Subramaia c a Departmet of Techology Maagemet, Faculty of Techology Maagemet ad Busiess, Uiversiti
More informationThe path polynomial of a complete graph
Electroic Joural of Liear Algebra Volume 10 Article 12 2003 The path polyomial of a complete graph C M da Foseca cmf@matucpt Follow this ad additioal wors at: http://repositoryuwyoedu/ela Recommeded Citatio
More informationApplied Mathematics Letters. On the properties of Lucas numbers with binomial coefficients
Applied Mathematics Letters 3 (1 68 7 Cotets lists available at ScieceDirect Applied Mathematics Letters joural homepage: wwwelseviercom/locate/aml O the properties of Lucas umbers with biomial coefficiets
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More information2.4 - Sequences and Series
2.4 - Sequeces ad Series Sequeces A sequece is a ordered list of elemets. Defiitio 1 A sequece is a fuctio from a subset of the set of itegers (usually either the set 80, 1, 2, 3,... < or the set 81, 2,
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s) that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationThe Adomian Polynomials and the New Modified Decomposition Method for BVPs of nonlinear ODEs
Mathematical Computatio March 015, Volume, Issue 1, PP.1 6 The Adomia Polyomials ad the New Modified Decompositio Method for BVPs of oliear ODEs Jusheg Dua # School of Scieces, Shaghai Istitute of Techology,
More informationSome families of generating functions for the multiple orthogonal polynomials associated with modified Bessel K-functions
J. Math. Aal. Appl. 297 2004 186 193 www.elsevier.com/locate/jmaa Some families of geeratig fuctios for the multiple orthogoal polyomials associated with modified Bessel K-fuctios M.A. Özarsla, A. Altı
More informationResearch Article Nonexistence of Homoclinic Solutions for a Class of Discrete Hamiltonian Systems
Abstract ad Applied Aalysis Volume 203, Article ID 39868, 6 pages http://dx.doi.org/0.55/203/39868 Research Article Noexistece of Homocliic Solutios for a Class of Discrete Hamiltoia Systems Xiaopig Wag
More informationAn Interpolation Process on Laguerre Polynomial
Global Joural of Pure ad Applied Mathematics. ISSN 0973-1768 Volume 13, Number 10 (2017), pp. 7089-7099 Research Idia Publicatios http://www.ripublicatio.com A Iterpolatio Process o Laguerre Polyomial
More informationProblem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient
Problem 4: Evaluate by egatig actually u-egatig its upper idex We ow that Biomial coefficiet r { where r is a real umber, is a iteger The above defiitio ca be recast i terms of factorials i the commo case
More informationOn a general q-identity
O a geeral -idetity Aimi Xu Istitute of Mathematics Zheiag Wali Uiversity Nigbo 3500, Chia xuaimi009@hotmailcom; xuaimi@zwueduc Submitted: Dec 2, 203; Accepted: Apr 24, 204; Published: May 9, 204 Mathematics
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More information