A survey on third and fourth kind of Chebyshev polynomials and their applications

详细信息    查看全文

• 作者：Kamal Aghigh ; M. Masjed-Jamei ; Mehdi Dehghan
• 关键词：Jacobi orthogonal polynomials ; Third and fourth kind of Chebyshev polynomials ; Weight function ; Hypergeometric functions ; Constant weighting factor
• 刊名：Applied Mathematics and Computation
• 出版年：2008
• 出版时间：15 May 2008
• 年：2008
• 卷：199
• 期：1
• 页码：2-12
• 全文大小：181 K

文摘

In this paper, we have a survey on Chebyshev polynomials of third and fourth kind, which are respectively orthogonal with respect to the weight functions f=""/science?_ob=MathURL&_method=retrieve&_udi=B6TY8-4PPW6XY-6&_mathId=mml21&_user=1067359&_cdi=5612&_rdoc=3&_acct=C000050221&_version=1&_userid=10&md5=fd2e34a1bc71927f0483c25358a035fa"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">ρ₁(x)=(1+x)^1/2(1-x)^-1/2 and f=""/science?_ob=MathURL&_method=retrieve&_udi=B6TY8-4PPW6XY-6&_mathId=mml22&_user=1067359&_cdi=5612&_rdoc=3&_acct=C000050221&_version=1&_userid=10&md5=784905c2639cb557df1c5d04a2d1e59a"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">ρ₂(x)=(1-x)^1/2(1+x)^-1/2 on [−1, 1]. These sequences are special cases of Jacobi polynomials f=""/science?_ob=MathURL&_method=retrieve&_udi=B6TY8-4PPW6XY-6&_mathId=mml23&_user=1067359&_cdi=5612&_rdoc=3&_acct=C000050221&_version=1&_userid=10&md5=ca279f0dba75a716d63d2b1617290d8c""> for f=""/science?_ob=MathURL&_method=retrieve&_udi=B6TY8-4PPW6XY-6&_mathId=mml24&_user=1067359&_cdi=5612&_rdoc=3&_acct=C000050221&_version=1&_userid=10&md5=655269a35ca5894e18337c444fd8a179"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">f"" alt=""greek small letter alpha"" title=""greek small letter alpha"" border=""0"">+β=0 and appear in the potential theory because of the nature of foresaid case differential equation. General properties of these two sequences such as orthogonality relations, differential equations, recurrence relations, decomposition of sequences, Rodrigues type formula, representation of polynomials in terms of hypergeometric functions, generating functions, their relation with the first and second kind of Chebyshev polynomials, upper and lower bounds and eventually estimation of two definite integrals as f=""/science?_ob=MathURL&_method=retrieve&_udi=B6TY8-4PPW6XY-6&_mathId=mml25&_user=1067359&_cdi=5612&_rdoc=3&_acct=C000050221&_version=1&_userid=10&md5=31b68399459e0c01932f18a75103f8a8""> and f=""/science?_ob=MathURL&_method=retrieve&_udi=B6TY8-4PPW6XY-6&_mathId=mml26&_user=1067359&_cdi=5612&_rdoc=3&_acct=C000050221&_version=1&_userid=10&md5=37814304e3f24644bc99d81271083f30""> are represented. Moreover, under the Dirikhlet conditions, an analytic function can be expanded in terms of the Chebyshev polynomials of third and fourth kind. Finally, what distinguishes these two sequences from other orthogonal polynomials is to satisfy a semi minimax property that has application in approximating the functions of type f=""/science?_ob=MathURL&_method=retrieve&_udi=B6TY8-4PPW6XY-6&_mathId=mml27&_user=1067359&_cdi=5612&_rdoc=3&_acct=C000050221&_version=1&_userid=10&md5=906e1055630925e5bfb392e65d380b9c"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">Q(x)P_n(x) where f=""/science?_ob=MathURL&_method=retrieve&_udi=B6TY8-4PPW6XY-6&_mathId=mml28&_user=1067359&_cdi=5612&_rdoc=3&_acct=C000050221&_version=1&_userid=10&md5=9c09282608904a8df4d384661614e139"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">P_n(x) is an arbitrary polynomial of degree n and f=""/science?_ob=MathURL&_method=retrieve&_udi=B6TY8-4PPW6XY-6&_mathId=mml29&_user=1067359&_cdi=5612&_rdoc=3&_acct=C000050221&_version=1&_userid=10&md5=f83ae1f1c627187639da228bf7f20971"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">Q(x) denotes a constant weighting factor. In this way, some numerical examples are also given.