----------------------- Page 1----------------------- AMBO UNIVERSITY SCHOOL OF GRADUTE STUDIES COLLEGE OF NATURAL AND COMPUTATIONAL SCIENCES DEPARTMENT OF MATHEMATICS Title: Bernoulli and Hermite Polynomials Associated with Jacobi Polynomials By Gena Husen Milki Adviser : Nasir Asfaw Kelifa (PhD) A Thesis Submitted to the Department of Mathematics in Partial Fulfillment of the Requirements for the Degree Masters of Science (M.Sc) in Mathematics (Analysis) Ambo University Ambo, Ethiopia December, 2022 ----------------------- Page 2----------------------- Declaration I, Gena Husen, here by declare that this thesis entitled "the relation between Bernoulli and Hermite associated with Jacobi polynomial" is the result of my own research work carried out by my self in a department of mathematics at Ambo University by the help of my advisor Dr.Nasir Asfawu. The work composed in the thesis has not been submitted to any where for the award of academic degree, diploma and certificate . |||||||||{ |||||||||||| |||||||||- Gena Husen Signature Date i ----------------------- Page 3----------------------- Approval sheet of the thesis Final approval and acceptance of the thesis is contingent upon the submission of the final copy of the thesis to the office of school of graduate studies through the departmental or school of graduate committee (DGC or SGC) of the candidate Name Signature Date Submitted by: |||||||||{ |||||||||||| |||||||||- Gena Husen Milki (PG Candidate) Approved by 1. Advisor: |||||||||{ |||||||||||| |||||||||- Nasir Asfaw Kelifa (PhD) Signature Date 2. H. Department Signature Date ||||||||||||- |||||||||||- ||||||||||{ 3. College/Institute Dean Signature Date ||||||||||| ||||||||||| |||||||||{ 4. School of Graduate Studies Signature Date |||||||||{ |||||||||||| |||||||||- ii ----------------------- Page 4----------------------- Certification AMBO UNIVERSITY SCHOOL OF GRADUATE STUDIES CERTIFICATION SHEET A thesis advisor, I have read and evaluated this thesis prepared under my guidance by Gena Husen Milki entitled \Bernoulli and Hermite polynomials associated with Jacobi polynomi- als". I recommend that it be submitted as fulfilling the thesis requirement. Nasir Asfaw (PhD ) |||||||| |||||||| Name of Advisor Signature Date As members of the board of examiners of the Msc thesis defense examined we certified that we have read and evaluated the thesis prepared by Gena Husen Milki and examined the candidate. We recommend that the thesis be accepted as fulfilling the thesis requirements for the degree of Master of Science in Mathematics (Analysis). 1. |||||| ||||||| ||||{ Chair Person Signature Date 2. ||||||{ ||||||| ||||{ Internal Examiner Signature Date 3. ||||||{ ||||||| ||||| External Examiner Signature Date iii ----------------------- Page 5----------------------- Dedication I dedicate this thesis to all my family members for their encouragement and enormous effort in my life and for success of my work. iv ----------------------- Page 6----------------------- Acknowledgments First of all I would like to thank my God who helped me through out my study and resist difficult time by giving me the strength and encouragement to complete this thesis. I express my special gratitude to my advisor Nasir Asfew(PhD) for his guidance and directory to overcome this work. I also thank the department mathematics of Ambo University for their hospitality during the period when this paper was written. v ----------------------- Page 7----------------------- Contents Declaration i Approval sheet of the thesis ii Certification iii Dedication iv Acknowledgments v Abstract 1 1 Introduction 2 1.1 Background of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Objective of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.1 General Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.2 Specific Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Significance of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Delimitation of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Review of Preliminary Concepts 6 2.1 Jacobi Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Bernoulli Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Bernoulli and Hermite Polynomials Associated with Jacobi Polynomials 18 3.1 Some Basic Definitions and Theorems . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Relationship between Bernoulli and Hermite polynomials associated with Ja- cobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 vi ----------------------- Page 8----------------------- 3.4 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References 24 vii ----------------------- Page 9----------------------- Abstract In 2017 N.U.Khan, T.Usman∗ and M.Aman [21] introduced the generating function for a Legendre based poly-Bernoulli polynomials and give some identities of these polynomials related to the stirling numbers of the second kind by making use of the generating function method and some functional equations. They conduct a further investigation in order to obtain some implicit summation formula for Legendre based poly-Bernoulli number and polynomials. In this thesis we study the relationship between Bernoulli and Hermite polynomial associated with Jacobi polynomials. The main purpose of this thesis is to investigate some identities of Bernoulli and Hermite polynomials associated with Jacobi polynomial arising from the orthogonality of Jacobi polynomials in the inner product space Pn . Finally the inner product space Pn of orthogonal Jacobi polynomial is an effective and used as a definitive modeling method to discuss relationship between Bernoulli and Hermite polynomials associated with Jacobi polynomials. 1 ----------------------- Page 10----------------------- Chapter 1 Introduction 1.1 Background of the Study The classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications in such areas as mathematical physics (in particular, the theory of random matrices), approximation theory, numerical analysis, and many others. Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien- Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, [4] the study of continued fractions to solve the moment problem by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials. The Bernoulli polynomials, [16] n   B (x) = X n B xn−k ; n k k k=0 named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler{MacLaurin formula. The first few Bernoulli polynomials are: B (x) = 1; 0 1 B (x) = x − ; 1 2 1 2 B (x) = x − x + ; 2 6 1 4 3 2 B (x) = x − 2x + x − ; 4 30 3 1 3 2 B (x) = x − x + x; 3 2 2 5 5 1 5 4 2 B (x) = x − x + x − x;::: 5 2 3 6 2 ----------------------- Page 11----------------------- These polynomials are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator) occur in the study of many special functions and, in particular, the Riemann zeta function and the Hurwitz zeta function. For the Bernoulli polynomials, the number of crossings of the x-axis in the unit interval does not go up with the degree. In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions. Bernoulli polynomials a similar set of polynomials, based on a generating function, is the family of Euler polynomials. Carl Gustav Jacob Jacobi (1804-1851) was a German mathematician active in many fields of mathematics. He is primarily remembered for his contributions to number theory and his work with elliptic functions. His Opuscula Mathematica (Collected Mathematical Works) was published in 1846. The Jacobi polynomials [7] are defined as P(α;β) (z) = (α + 1)n F (−n; 1 + α + β + n; α + 1; 1 − z) n n! 2 1 2 where (α) is Pochhammer’s symbol (for the rising factorial) given by (α) = (α)(α+ 1)(α+ n n 2):::(α + n − 1) and F is a Guass hypergeometric function given by: 2 1 1 X (α ) (α ) 1 n 2 n F (α ;α ;β ; z) = : 2 1 1 2 1 (β ) n! 1 n n=0 In this case, the series for the hypergeometric function is finite, also known as hyper ge- ometric polynomials, they are solutions to the Jacobi differential equation, and give some other special named polynomials as special cases. They are implemented in the Wolfram Language as Jacobi P[n, a, b, z]. Charles Hermite, [11] was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian op- erators, and cubic Hermite splines are named in his honor. The Hermite polynomial H (x) n is defined by the formula 2 n −x 2 d e n x H (x) = (−1) e ; n = 0;1;2;::: n dxn The first few Hermite polynomials are: H (x) = 1; 0 H (x) = 2x; 1 2 H (x) = 4x − 2; 2 3 ----------------------- Page 12----------------------- 3 H (x) = 8x − 12x; 3 4 2 H (x) = 16x − 48x + 12::: 4 This permits us to study identities of Bernoulli and Hermite polynomials associated with Jacobi polynomials. 1.2 Statement of the Problem In this paper, we consider a new class of generating functions for Hermite-Bernoulli-Jacobi polynomials and study certain implicit summation formulas by using different analytical means and applying generating function. We also consider bilateral series associated with a newly-introduced generating function by appropriately specializing a number of known or new partly unilateral and partly bilateral generating functions. The results presented here, being very general, are pointed out to be specialized to yield a number of identities involving relatively simpler and familiar polynomials. this study will attempt to answer the following questions: 1. How can we apply the orthogonality property of Hermite and Jacobi polynomials to investigate their relationship with Bernoulli polynomials? 2. Is it possible to identify identities on the Bernoulli and Hermite polynomials arising from the orthogonality of Hermite and Jacobi polynomials in the inner product space P ? n 3. How can we find Bernoulli, Hermite and Jacobi polynomials identities by using their generating functions? 1.3 Objective of the Study 1.3.1 General Objective The general objective of this study is to investigate some identities of Bernoulli and Hermite polynomials associated with Jacobi polynomial by applying the orthogonality property and generating functions of the polynomials. 4 ----------------------- Page 13----------------------- 1.3.2 Specific Objectives The specific objectives of the study are: 1. To apply properties of orthogonality of Hermite and Jacobi polynomials and relate with Bernoulli polynomials. 2. To discuss the properties of Bernoulli, Hermite and Jacobi polynomials and some relations between the properties of these polynomials. 3. To investigate some identities of Bernoulli and Hermite associated with Jacobi poly- nomial by proving theorems and giving examples. 1.4 Significance of the Study This study will be considered to have vital importance for the following reasons: 1. The results obtained will contribute new concept to property of Bernoulli Hermite and associated Jacobi polynomial 2. It investigates some relation ship of orthogonal polynomials Hermite and Jacobi poly- nomial with non orthogonal Bernoulli polynomial . 3. It will be used as reference material for anyone who works on the area. 1.5 Delimitation of the Study In this study we focus on the inner product space of Jacobi polynomials to discuss the relationship of Bernoulli and Hermite polynomials associated with Jacobi polynomials. 5 ----------------------- Page 14----------------------- Chapter 2 Review of Preliminary Concepts Recently much attention was drawn to Bernoulli and Hermite associated with Jacobi Poly- nomial. In 2012, Kim et al, [1] Investigated some identities on the Bernoulli and the Hermite polynomials arising from the orthogonality of Jacobi polynomials in the inner product space P n In 2019,[12] in the classical connection problem, it is dealt with determining the coefficients in the expansion of the product of two polynomials with regard to any given sequence of polynomials. As a generalization of this problem, they will considered sums of finite prod- ucts of Fubini polynomials and represent these in terms of orthogonal polynomials. Here, the involved orthogonal polynomials are Chebyshev polynomials of the first, second, third and fourth kinds, and Hermite, extended Laguerre, Legendre, Gegenbauer, and Jabcobi polyno- mials. These representations are obtained by explicit computations. \In 2019 Taekyun Kim 1 et al, [6] represented sums of finite products of Legendre and La- guerre polynomials in terms of several orthogonal polynomials." Indeed, by explicit compu- tations they expressed each of them as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, some of which involve terminating hyper ge- ometric functions. In 2019 Taekyun Kim 1 et al, [3] represented sums of finite products of Legendre and Laguerre polynomials in terms of several orthogonal polynomials. Indeed, by explicit computations they expressed each of them as linear combinations of Hermite, generalized Laguerre, Legen- dre, Gegenbauer and Jacobi polynomials, some of which involve terminating hyper geometric functions F and F . 1 1 2 1 In 2013 Chen et al, [13] introduced and investigated an extension of the generalized Apostol- Euler polynomials. They stated some properties for these polynomials and obtain some relationships between the polynomials and Apostol-Bernoulli polynomials, Stirling numbers of the second kind, Jacobi polynomials, Laguerre polynomials, Hermite polynomials and generalized Bernoulli polynomials. In 2019 Shen, Shimeng and Chen, Li, [13] firstly introduced a new non-linear recursive se- 6 ----------------------- Page 15----------------------- quence . Then, using this sequence, a computational problem involving the convolution of the Legendre polynomial is studied using the basic and combinatorial methods. Finally, they gave an interesting identity. In 2013 Araci, Serkan et.all, [17] deal mainly with arithmetic properties of Legendre poly- nomials by using their orthogonality property. they showed that Legendre polynomials are proportional with Bernoulli, Euler,Hermite and Bernstein polynomials. In 2009 Balderrama, Cristina, [18] were constructed and studied families of generalized or- thogonal polynomials with hermitian matrix argument associated to a family of orthogonal polynomials on R. Different normalizations for these polynomials are considered and they obtain some classical formulas for orthogonal polynomials from the corresponding formulas for the one dimensional polynomials. they also constructed semi groups of operators associated to the generalized orthogonal polynomials and they gave an expression of the infinitesimal generator of this semi group and in the classical cases, we prove that this semi group is also Markov. For d{dimensional Jacobi expansions we study the notions of fractional integral (Riesz poten- tials), Bessel potentials and fractional derivatives. They presented a novel decomposition of the L space associated with the d{dimensional Jacobi measure and obtain an analogous of 2 Meyer’s multiplier theorem in this setting. Sobolev Jacobi spaces are also studied. 2.1 Jacobi Polynomials Definition 2.1. Let α;β 2 R with α> −1 and β > −1, the Jacobi polynomials P(α;β) (z) n are defined as (α;β) (α + 1)n  1 − z P (z) = F −n; 1 + α + β + n; α + 1; n n! 2 1 2 n   k (2.1) n X (α + 1)n k (1 + α + β + n) z − 1 = ; n! (α + 1)k 2 k=0 where (α) =α(α + 1):::(α + n − 1) = Γ(α + n)=Γ(α) and F is a Guass hypergeometric n 2 1 function given by: 1 X (α ) (α ) 1 n 2 n F (α ;α ; β ; z) = : 2 1 1 2 1 (β ) n! 1 n n=0 From(2.1), we note that n   k n X P(α;β) (z) = Γ(α + n − 1) k Γ(α + β + n + k + 1) x − 1 : (2.2) n n!Γ(α + β + n + 1) Γ(α + k + 1) 2 k=0 7 ----------------------- Page 16----------------------- By (2.2), we see that P(α;β) (z) is a polynomial of degree n with real coefficients. It is not n (α;β)  difficult to show that the leading coefficient of Pn (z) is 2−n α+β+2n . From (2.2), we have n (α;β)  Pn (1) = α+n n By (2.1), we get d k P(α;β) (z) = 2k Γ(n + α + β + k + 1)p(α+k;β+k) (z); (2.3) dz n Γ(n + α + β + 1) n−k where k is a positive integer. The Rodrigues’ formula for P(α;β) (z) is given by n α β (α;β) (−1)n d k n+α n+β (1 − z) (1 + z) Pn (z) = n ( ) (1 − z) (1 + z) : (2.4) 2 n! dz It is easy to show that u = P(α;β) (z) is a solution of the following differential equation : n 2 00 0 (1 − x )u + β − α − (α + β + 2)zu + n(n + α + β + 1)u = 0: (2.5) As is well known, the generating function of P(α;β) (z) is given by n 1 α+β X 2 (α;β) n F(z;t) = n=0 Pn (z)t = R(1 − t + R)α (1 + t + R)β ; (2.6) where R = p 2 1 − 2zt + t From (2.3),(2.4) and (2.6), we can drive the following identity Z 1 (α;β) (α;β) α β 2α+β+1Γ(n + α + 1)(n + β + 1) Pm (z)Pn (1 − z) (1 + z) dz = σn;m ; −1 (2n + α + β + 1)Γ(n + α + β + 1)Γ(n + 1) (2.7) where σn;m is Kronecker symbol and 8 > σn;m = <1 ifm = n : (2.8) > : 0 ifm =6 n: Let Pn = fp(z) 2 R bzc jdeg p(z)j ≤ ng then Pn is an inner product space with respect to the inner product Z 1 α β hq (z);q (z) = (1 − z) (1 + z) q (z)q (z)dz 1 2 1 2 −1 where q (z);q (z) 2 P . 1 2 n From (2.7), we note that P(α;β) (z);P(α;β) (z):::;P(α;β) (z) is an orthogonal basis for P . 0 1 n n 8 ----------------------- Page 17----------------------- Definition 2.2. Recurrence Formula The Jacobi polynomial are generated by three -term recurrence relation α;β α;β α;β α;β α;β α;β (x):n ≥ 1 (2.9) P = (a − b )J (x) = C Jn−1 n n n 1 1 Pα;β (x) = 1;Pα;β (x) = (α + β + 2) + (α − β); 0 1 2 2 where α;β (2n + α + β + 1)(2n + α + β + 2) an = (2.10) 2(n + 1)(n + α + β + 1) 2 2 bα;β = (β − α )(2n + α + β + 1) (2.11) n 2(n + 1)(n + α + β + 1)(2n + α + β) α;β (n + α)(n + β)(2n + α + β + 2) cn = (2.12) (n + 1)(n + α + β + 1)(2n + α + β) Theorem 2.1. Rodrigues’ formula The Rodrigues’ formula for the Jacobi polynomial is stated below n α β α;β (−1) dn n+α n+β (1 − x) (1 + x) Pn (x) = n n [(1 − x) (1 + x) ]: (2.13) 2 n!dx Proof. For any 2 pn−1 .using integration by parts leads to Z 1 @n (1 − x)n + α(1 + x)n+β  dx x −1 n Z 1  n+β  n = (−1) (1 − x)n + α(1 + x) @ d = 0 x x −1 Hence there exist a constant c such that n n n+β β α;β @ ((1 − x)n + α(1 + x) ) = c (1 − x)α(1 + x) P (x); x n n letting x ! 1 leads to c = 1 1  n+β  n n n @ (1 − x)n + α(1 + x) = (−1) n!2 : Pα;β (1) (1 − x)α(1 + x)β x n th We now present some consequences of the (2.13).First ,expanding the n -order derivative of (2.13) yields the explicit formula. n    Pα;β (x) = 2−n X n + α n + β (x − 1)n−p (x + 1)p (2.14) n p n − p p=0 9 ----------------------- Page 18----------------------- Second replacing x by -x in (2.13)immediately leads to the symmetric relation Pα;β n α;β n (−x) = (−1) Pn (x) (2.15) Therefore , the special Jacobi polynomial Pα;β ,is an odd function for odd n and an even n function for even n. From the above property we have α;β n Γ(n + β + 1) Pn (−1) = (−1) (2.16) n!Γ(β + 1) This relation allows us to evaluate the Jacobi polynomial at any given abscissa x 2 [−1;1]. 2.2 Bernoulli Polynomials Definition 2.3. The Bernoulli polynomials B (x) defined by: n n   X n k B (x) = Bn−kx : (2.17) n k k=0 Pn  n k The Bernoulli polynomials B (x) = B x , named after Jacob Bernoulli, combine n k=0 k n−k the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler{MacLaurin formula. Definition 2.4. A Reccurrecnce formula The Bernoulli numbers B are defined recursively as: n   n−1 B = 1;andX n B = 0 for n = 2;3;4::: (2.18) 0 k k k=0 The recursive formula (2.18) generates an interesting sequence of rational numbers B , the n sequence of Bernoulli numbers. The sequence of Bernoulli numbers B possess several inter- n esting properties. By using formula (2.18) we list the first few Bernoulli numbers as −1 1 B = ; B = ; B = 0; 1 2 3 2 6 −1 1 −1 5 B = ;B = 0;B = ;B = 0;B = ;B = 0;B ;::: 4 5 6 7 8 9 10 30 42 30 66 0 Observe that the B s with odd indexes are vanishing terms. This is one of the properties n of B .Which Bernoulli number posses and we discuss several other properties of Bernoulli n 10 ----------------------- Page 19----------------------- numbers latter. This is one of the properties of B which Bernoulli numbers possess and we n discuss several other properties of Bernoulli numbers later. Jacob Bernoulli considered the sequence B (x) of Bernoulli polynomials; they can be defined by a recursive formula, or by n n   B (x) = X n B xn−k (2.19) n k k k=0 Definition 2.5. The generating functions Bernoulli polynomials are also defined by a generating function as 1 xz n ze = XB (x) z (2.20) n z e − 1 n! n=0 If we put x = 0 in (2.20), we get the Bernoulli numbers B = B (0). That is, the Bernoulli n n numbers Bn are defined by generating function as: 1 n z z = XBn z (2.21) e − 1 n! n=0 Theorem 2.2. Appell sequence We have an other alternative approach towards the definition of Bernoulli polynomials and numbers. The following is an appell sequence of B (x), given by n B (x) = 1 (2.22) 0 0 B (x) = nBn−1 (x) (2.23) n 8 Z > < 1 1; n = 0 B (x)d = (2.24) n x 0 > : 0; n> 0 Proof. By differentiating equation (2.17) we get n   X d n 0 k B (x) = Bn−kx n dx k k=0 n   = X n kBn−kxk−1 k k=1   n−1 = X n (k + 1)Bn−(k+1)xk k + 1 k=0   n−1 X n − 1 = n Bn−1−kxk k k=0 0 B (x) = nBn−1 (x); n   where we use (k + 1) n = n n−1 k+1 k 11 ----------------------- Page 20----------------------- Each of the the definitions, that is, the recurrence formula (2.19), the generating func- tion(2.20) and Appell sequence ((2.22),(2.23),(2.24)), generate the same sequence of polyno- mials . Some of the well known properties of Bernoulli polynomials are : • Symmetry property n B (1 − x) = (−1) B (x) for n ≥ 0 (2.25) n n • Difference equation B (x + 1) − B (x) = nxn−1 for n ≥ 1 (2.26) n n • Addition formula n   B (x + y) = X n B (x)yn−1 (2.27) n k k k=0 • Raabe’s multiplication m−1 X k B (mx) = mn−1 B (x + ) (2.28) n n m k=0 where m and n are integers with n ≥ 0 and m ≥ 1: k Bernoulli polynomials are expressed in terms of the canonical basis(x )k2N of R[x]. In fact, we have proved that n   B (x + y) = X n Bn−k (x)yk (2.29) n k k=0 k and this can be used to, conversely, express the canonical basis (x )k2N of R[x] in terms of Bernoulli polynomials.Indeed, substituting y= 1 in (2.29) we obtain n   B (x + 1) = X n B (x); n n−k k k=0 but according to (2.26) we have also n Bn+1 (x + 1) = Bn+1 (x) + (n + 1)x : Thus n+1   X n + 1 n (n + 1)x = Bn+1−k (x) k k=1 n   X n + 1 = B (x) k k k=0 Finally n   X 1 n + 1 n x = B (x): k n + 1 k k=0 12 ----------------------- Page 21----------------------- 2.3 Hermite Polynomials Definition 2.6. The Hermite polynomial H (x) can be defined by the formula n 2 n −x 2 d e n x H (x) = (−1) e ; n = 0;1;2;::: n dxn Definition 2.7. Generating Function The Hermite polynomial may be summed to yield the generating function 1 n w(x;z) = e−z2 +2zx = XH (x) z : (2.30) n n! n=0 The special Values of the Hermite polynomials follow from the generating function 1 n 1 2n w(0;z) = e−z2 = XH (0)z = X(−1)n z : n n! n! n=0 n=0 A comparison of coefficients of these power series yields 8 > n (2n)! : H2n+1 (0) = 0: Similarly we obtain from the generating function identity −z2 −2zx w(−x;z) = e w(x; −z): (2.32) The power series identity 1 1 n n XH (−x) t = XH (x) (−t) : (2.33) n n! n n! n=0 n=0 The first few Hermite polynomials are H (x) = 1; 0 H (x) = 2x; 1 2 H (x) = 4x − 2; 2 H (x) = 8x3 − 12x; 3 4 2 H (x) = 16x − 48x + 12::: 4 13 ----------------------- Page 22----------------------- Definition 2.8. Explicit representation bn c 2 n n−2k H (x) = X (−1) n!(2x) ; (2.34) n k!(n − 2k)! k=0 where bn c denotes the largest integer ≤ n 2 2 The Hermite polynomials (or more exactly, the Hermite polynomials multiplied by the con- stant factor 1 ) are the coefficients in the expansion n! 1 2 XH (x) w(x;z) = e2xz −z n n = z ; jzj < 1 (2.35) n! n=0 and hence w(x;z) is called the generating function of the Hermite polynomials. The prove of (2.35) is found in as follows; We need only note that z, is an entire function, and therefore has the Taylor series 1  n  X 2 1 @ w 2xz −z n w(x;z) = e = z ; jzj < 1 n! @zn n=0 z=0 which immidiately implies (2.35), since  n   n  @ w 2 @ 2 x −(x−z) = e e @zn @zn z=0 z=0 " n −u2 # 2 d e n x = (−1) e dun u=x = H (x) n Formula (2.35) can be used to drive various properties of the Hermite polynomials. For 2 example, setting x = 0 in (2.37), expanding e−z in power series, and comparing coefficients of powers of z in both sides of the resulting equation, we find that n (2n)! H (0) = (−1) ; H (0) = 0 (2.36) 2n 2n+1 n! Recurrence Relations and Differential Equation for the Hermite Polynomials • 2xH (x) = 2nH (x) + H (x) n n−1 n+1 14 ----------------------- Page 23----------------------- 0 • H (x) = 2nHn−1 (x) + Hn+1 (x) n 0 0 • xH (x) = nH (x) + nH (x) n n−1 n 0 • H (x) = 2xH (x) − H (x) n n n+1 Substituting (2.35) into the identity @w − (2x − 2z)w = 0 @z (a power series can always be differentiated term by term), we find that 1 1 1 XH (x) XH (x) XH (x) n+1 n n n n n+1 z − 2x z + 2 z = 0; n! n! n! n=0 n=0 n=0 which gives H (x) − 2xH (x) + 2nH (x) = 0; n = 1;2;3;::: (2.37) n+1 n n−1 n when the coefficient of z is equated to zero. The recurrence relation (2.37) connecting three Hermite polynomials with consecutive indices, can be used to calculate the Hermite polyno- mials step by step, starting from H (x) = 1; H (x) = 2x: 0 1 We can drive another recurrence relation satisfied by the Hermite polynomials by substitut- ing (2.35) into the identity @w − 2zw = 0: @z This gives 1 0 1 XH (x) XH (x) n n n n+1 z − 2 z = 0; n! n! n=0 n=0 or 0 H (x) = 2nHn−1 (x); n = 1;2;::: (2.38) n Formula (2.38) allows us to express the derivative of a Hermite polynomial in terms of an- other Hermite polynomial, and is very useful. Using the recurrence relations(2.37 and 2.38), we can easily drive a differential equation satisfied by the Hermite polynomials. In fact, eliminating Hn−1 (x) from these two relations, we obtain 15 ----------------------- Page 24----------------------- H (x) − 2xH (x) + H0 (x) = 0: n+1 n n Then, differentiating this formula and using (2.38) again, we find that H00 (x) − 2xH0 (x) + 2nH (x) = 0; n = 0;1;2;::: (2.39) n+1 n n Theorem 2.3. Orthogonality property of Hermite polynomial Z 1 −x2 n pπσ (2.40) H (x)H (x)e dx = 2 n! n;m n m −1 wher σn;m is defined in equation (2.8) Proof. We know that the generating function for Hermite polynomial is 1 n e−z2 +2zx = XH (x) z (2.41) n n! n=0 similarly 1 m e−s2 +2sx = XH (x) s (2.42) m m! m=0 multiplying equation(2.41) and equation (2.42) gives 1 1 n m e−z2 +2zx :e−s2 +2sx = XH (x) z :XH (x) s (2.43) n n! m m! n=0 m=0 2 multiply equation(2.43) by e−x gives 1 1 n m e−x2 e−z2 +2zx :e−s2 +2sx = e−x2 XH (x) z :XH (x) s (2.44) n n! m m! n=0 m=0 integrate equation (2.44) from −1 to 1 gives Z Z 1 1 1 1 n m e−x2 e−z2 +2zx :e−s2 +2sxdx = e−x2 XH (x) z :XH (x) s n n! m m! −1 −1 n=0 m=0 if m=n Z Z 1 1 1 1 n n e−x2 e−z2 +2zx :e−s2 +2sxdx = e−x2 XH (x) z :XH (x) s n n! n n! −1 −1 n=0 n=0 Z Z 1 1 1 n n −x2 −s2 +2sx−z2 +2zx −x2 X 2 z s e dx = e H (x) dx −1 −1 n=0 n (n!)2 Z Z 1 1 1 n n e−(x−s−z)2 :e2zs = e−x2 XH2 (x) z s dx −1 −1 n=0 n (n!)2 16 ----------------------- Page 25----------------------- Z 1 n 1 n n pπ(2zs) = e−x2 XH2 (x) z s dx n! −1 n=0 n (n!)2 Z 1 n n 1 p (2zs) X (zs) −x2 2 π = e H (x)dx n! n=0 (n!)2 −1 n Z 1 1 n 2 e−x2 X 2 pπ(2zs) :(n!) H (x)dx = −1 n=0 n n! (zs)n Z 1 1 e−x2 X 2 n pπ H (x)dx = n!2 n −1 n=0 17 ----------------------- Page 26----------------------- Chapter 3 Bernoulli and Hermite Polynomials As- sociated with Jacobi Polynomials 3.1 Some Basic Definitions and Theorems Definition 3.1. Inner product space An inner product on the normed space X over C is a map X ! X ! C satisfying x,y 2 X; 1. hx + y;zi = hx;zi + hy;zi for all x,y,z 2 X: 2. hy;xi = hx;yi for all x,y 2 X. 3. hαx;yi = αhx;yi for all x,y 2 X and α 2 C. 4. For any x 2 X; hx;xi 2 R;hx;xi ≥ 0 and hx;xi =6 0 if x =6 0: The pair (X; h:;:i) is called in inner product space. Definition 3.2. Gamma function The Gamma function is defined by the formula Z 1 −t z −1 Γ(z) = e t dt; <(z) > 0 0 whenever the complex variable z has a positive Real part <(z) we can write Γ(z) as a sum of two integrals Z 1 −t z −1 Z 1 −t z −1 Γ(z) = e t dt + e t dt: 0 1 Where it can be easily be shown that the first integral defines a function p(z) which is analytic in the half plane <(z) > 0; while the second integral define an entire function. It follows that function Γ(z) = P(z) + Q(z) is analytic in the half plane <(z) > 0: Some relations satisfied by Γ function are: Γ(z + 1) = zΓ(z) 18 ----------------------- Page 27----------------------- π Γ(z)Γ(1 − z) = sin πz p 1 22z −1Γ(z)Γ(z + ) = πΓ(2z): 2 3.2 Relationship between Bernoulli and Hermite poly- nomials associated with Jacobi polynomials In this section we will investigate some relationship between Bernoulli and Hermite associated with Jacobi polynomials arising from the inner product space of Jacobi polynomials. Our main result stated as follows. From (2.4), we have n       k n+k Pα;β (z) = X n + α n + β z − 1 z + 1 (3.1) n n − k k 2 2 k=0 By (2.1),we have I 1 n+α n+β XPα;β (x) = 1 (1 + ((x + 1)=2)z) (1 + ((x − 1)=2)z) dz (3.2) n 2πi zn+1 n=0 where we assume x =6 ±1 and circle around 0 is taken so small that −2(x ± 1)−1 lie neither on it nor in its interior. It is not so difficult to show that Pα;β n α;β (−x) = (−1) P P (x): n n Proposition 3.1. For q(x) 2 P , we have n n q(x) = XC Pα;β (x);(C 2 R) (3.3) k n k k=0 where k Z 1  k  (−1) (2k + α + β + 1)Γ(k + α + β + 1) d k+α k+β C = (1 − x) (1 + x) q(x)dx: k 2α+β+1+k Γ(α + k + 1)Γ(β + k + 1) −1 dxk (3.4) 19 ----------------------- Page 28----------------------- n Proof Let us take q(x) = x 2 P : First, we consider the following integral: n Z 1 d k (1 − x)k+α (1 + x)k+β q(x)dx −1 dx = Z 1 ( d k k+α k+β n ) (1 − x) (1 + x) x dx −1 dx = (−n) Z 1 d k−1 (1 − x)k+α (1 + x)k+β xn−1dx −1 dx = ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: = (−1)n n! Z 1 (1 − x)k+α (1 + x)k+β xn−k dx (3.5) (n − k)! −1 k 2k+α+β+1 Z 1 (−1) n!2 k+β k+α n−k = y (1 − y) (2y − 1) dy (n − k)! 0   k n−k X (−1) n! 2k+α+β+1 n − k l n−k−1 = 2 2 (−1) B(k + l + β + 1;k + α + 1) (n − k)! l l=0   k n−k = (−1) n! 22k+α+β+1 X n − k l n−k−1 Γ(k + l + β + 1)Γ(k + α + 1) 2 (−1) (n − k)! l Γ(2k + α + β + l + 2) l=0 By Equation (3.4) and (2.6) we have k Z 1 d (−1) (2k + α + β + 1)Γ(k + α + β + 1) k k+α k+β n C = x ( ) (1 − x) (1 + x) x dx k 2α+k+1Γ(α + k + 1)Γ(β + k + 1) −1 dx k k (−1) (2k + α)Γ(k + α + β + 1) (−1) n!22k + α + β + 1 = : 2α+β+k+1 (n − k)!   n−k X n − k l n−k−l Γ(k + l + β + 1)Γ(k + α + 1) × 2 (−1) l Γ(2k + α + β + l + 2) l=0   k n−k l = (2k + α + β + 1)Γ(k + α + β + 1)n!2 :X(−1)n−k−l n − k 2 Γ(k + l + β + 1) Γ(β + k + 1)(n − k)! l Γ(2k + α + β + l + 2) l=0 (3.6) ByProposition (3.1), we get   n n−1 n XX (2k + α + β + 1)Γ(2k + α + β + 1) k x = n! 2 Γ(k + α + 1)(n − k)! k=0 l=0 ! (3.7)  n−k−1 n−l l (−1) l 2 Γ(k + l + β + 1) α;β × Pk (x) Γ(2k + α + β + l + 2) From (2.17), we have 1   n xt 1 t xt t X Bn+1 (x + 1) − Bn+1 (x) t e = t e (e − 1) = : (3.8) t e − 1 n + 1 n! n=0 20 ----------------------- Page 29----------------------- By (3.8),we get n Bn+1 (x + 1) − Bn+1 (x) x = ;(n 2 Z+ ) (3.9) n + 1 There fore by(2.17) and (3.9), we obtain the following theorem Theorem 3.1. For n 2 Z+ , one has  ! n n−k n−k−l k+l n−k X X 1 (−1) 2 (2k + α + β + 1) l Bn+1 (x + 1) − Bn+1 (x) = (n + 1)! (k + β + 1)Γ(2k + α + β + l + 2)(n − k)! k=0 l=0 α;β ×Γ(k + α + β + 1)Γ(k + l + β + 1)p (x): k (3.10) Proof. Let us take q(x) = B (x) 2 p .Then we evaluate the following integral n n Z 1 ( d k k+α k+β ) (1 − x) (1 + x) B (x)dx −1 dx n n   k Z 1 = X nl Bn−1 (−1) l! 22k+α+β+1 yk+β (2y − 1)l−k dy (l − k)! 0 l=k     n k l−k = X n Bn−1 (−1) l! 22k+α+β+1 X l − k 2m (−1)l−k−m (3.11) l=k l (l − k)! m=0 m Γ(k + m + β + 1)Γ(k + 2 + 1) × Γ(2k + α + β + m + 2) n l−k   n ( l−m 2k+α+β+1 l−k m XX l Bn−1 (−1) − 1) l!2 m 2 Γ(k + m + β + 1)Γ(k + α + 1) = l=k m=0 (l − k)!Γ(2k + α + β + m + 2) finding (3.4) and (2.1), we have Theorem 3.2. For n 2 Z+ , one has n n l−k  ! k+m n l−k l−k X XX 2 l Bn−1 (−1) l!(2k + α + β + 1) m B (x) = n k=0 l=k m=0 Γ(β + k + 1)(l − k)!Γ(2k + α + β + m + 2) (3.12) ×Γ(k + m + β + 1)Γ(k + α + β + 1)Pα;β (x) k Proof. Let q(x) = Pα;β (x) 2 P . n n From Proposition (3.1), we firstly evaluate the following integral Z 1 d k k+α k+β α;β ( ) (1 − x) (1 + x) p (x)dx −1 dx n (3.13) = (−1)k 1 γ(n + α + β + k + 1) Z 1 (1 − x)k+α (1 + x)k+βp(α+β+k) (x)dx 2k Γ(n + α + β + 1) −1 n−k 21 ----------------------- Page 30----------------------- By (3.1) and (3.12) , we get Z 1 d k k+α k+β α;β ( ) (1 − x) (1 + x) p (x)dx = −1 dx n    k l−k (−1) Γ(n + α + β + k + 1) X n + α n + β k 2 Γ(n + α + β + 1) n − k − l l l=0 ×Z0 1 (1 − y)k+α+1yn+β −l dy =    n−k k α+β+k+1 Γ(n + α + β + k + 1) X n + α n + β (−1)l (−1) 2 Γ(n + α + β + y) n − k − l l l=0 ×B(k + α + l + 1;n+ β − l + 1) =    n−k k α+β+k+1 Γ(n + α + β + k + 1) X n + α n + β l (−1) 2 (−1) Γ(n + α + β + 1) n − k − l l l=0 Γ(α + k + l + 1)Γ(n + β − l + 1) × Γ(α + β + k + n + 2) 3.3 Summary and Conclusion 3.3.1 Summary In this thesis, we used inner product of Jacobi Polynomials to investigate some relation- ships between Bernoulli and Hermite polynomials associated with Jacobi polynomials. We first gives the definitions of Bernoulli, Hermite and Jacobi polynomials and also we defined Bernoulli and Hermite polynomials by using their generating functions and we discussed the inner product of Jacobi polynomials. Also we have given some formula of Bernoulli and Hermite polynomials related to the Jacobi polynomials by using different analytical means and applying the generating functions Bernoulli and Hermite polynomials and the inner product of Jacobi polynomials in section 3.2. These results investigates some relationships between Bernoulli and Hermite polynomials associated with Jacobi polynomials arising from the inner product of Jacobi polynomials. In general to sum up this the relationship between Bernoulli and Hermite polynomial associated with Jacobi polynomials is proved by applying the inner product of Jacobi polynomial discussed under section 2.1 with analytical method subjected with the generating functions of Bernoulli and Hermite polynomials. 22 ----------------------- Page 31----------------------- 3.3.2 Conclusion The main intension of this thesis is to investigate some relationships between Bernoulli and Hermite polynomials associated with Jacobi polynomials arising from the inner product of Jacobi polynomials. So in this thesis, we studied about Bernoulli, Hermite and Jacobi polynomials specifically concerning with the relationship between Bernoulli and Hermite associated with Jacobi polynomials arising from the inner product of Jacobi polynomials as well as we introduced and explained definitions of Bernoulli, Hermite and Jacobi polynomials, some preliminaries that used to study the relationships between those polynomials and the combinations of Generating functions of Bernoulli and Hermite polynomials using the inner product of Jacobi polynomials. This relationship were showed and proved by applying very effective method and power full mathematical tool called the inner product space of Jacobi polynomials. There fore, this relationship of Bernoulli and Hermite polynomials associated with Jacobi polynomials were studied in this research supported with theorems and propositions. In general in this research we investigated some relationships between Bernoulli and Hermite polynomials associated with Jacobi polynomials arising from the inner product of Jacobi polynomials. So inner product space of Jacobi polynomials is very interesting and and effective method as well as the easiest and a definitive modeling method to show the relationship between Bernoulli and Hermite polynomials associated with Jacobi polynomials. 3.4 Recommendation The main important issues for the future work regarding to investigate some relationships between Bernoulli and Hermite associated with Jacobi polynomials are: 1. Depending on this result and try to improve more approaches which help to show the relationship between Bernoulli and Hermite associated with Jacobi polynomials. 2. Apply the inner product space of Jacobi polynomials to investigate the relationship between some other polynomials associated with Jacobi polynomials. 3. I recommend that further researchers able to find more appropriate methods to in- 23 ----------------------- Page 32----------------------- vestigate some identities of Bernoulli and Hermite polynomials associated with Jacobi polynomials in order to reduce difficulties in such types of polynomials and extend this method to investigate some identities of Bernoulli and Hermite associated with Jacobi polynomials. 24 ----------------------- Page 33----------------------- References [1] Kim, Taekyun,Some identities on Bernoulli and Hermite polynomials associated with Jacobi polynomials, Discrete Dynamics in Nature and Society (2012). [2] Kim, Dae San and Dolgy,Representing by orthogonal polynomials for sums of finite products of Fubini polynomials Mathematics7(2019),no.4,319. [3] Kim, Taekyun and Hwang,Connection problem for sums of finite products of Legendre and Laguerre polynomials Symmetry 11(2019),no.3,317. 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