Exceptional Meixner and Laguerre orthogonal polynomials

详细信息    查看全文

• 作者：Antonio J. Durá ; n ^{duran@us.es" class="auth_mail}
• 关键词：Orthogonal polynomials ; Exceptional orthogonal polynomial ; Difference operators ; Differential operators ; Meixner polynomials ; Krawtchouk polynomials ; Laguerre polynomials
• 刊名：Journal of Approximation Theory
• 出版年：August 2014
• 年：2014
• 卷：184
• 期：Complete
• 页码：176-208
• 全文大小：363 K

文摘

Using Casorati determinants of Meixner polynomials f&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=047e45d4f362c940c55ec1fdc66ff6ad">f" data-inlimgeid="1-s2.0-S0021904514000938-si1.gif">flow="scroll">(mna,c)n, we construct for each pair formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000938&_mathId=si2.gif&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=9b51ed7a58678b8a4d282ac40a742250" title="Click to view the MathML source">F=(F₁,F₂)flow="scroll">F=(F1,F2) of finite sets of positive integers a sequence of polynomials f&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=82039bc3b6f73045cecf523cea4a1f68">f" data-inlimgeid="1-s2.0-S0021904514000938-si3.gif">flow="scroll">mna,c;F, formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000938&_mathId=si4.gif&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=233606fa2d2b74c181287a92670db73f" title="Click to view the MathML source">n∈蟽_Fflow="scroll">n∈蟽F, which are eigenfunctions of a second order difference operator, where formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000938&_mathId=si5.gif&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=bb9280499f52c1e20b57d90a6a2f1a03" title="Click to view the MathML source">蟽_Fflow="scroll">蟽F is certain infinite set of nonnegative integers, formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000938&_mathId=si6.gif&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=337baaaf9862441ce9859ecbdb92d856" title="Click to view the MathML source">蟽_F鈯婏竴Nflow="scroll">蟽F鈯婏竴N. When formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000938&_mathId=si7.gif&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=c0f33bf725689cc3a66d4ed786ee66f5" title="Click to view the MathML source">cflow="scroll">c and formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000938&_mathId=si8.gif&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=af75ab2ddfe7f4e3563f2d1f034b4d19" title="Click to view the MathML source">Fflow="scroll">F satisfy a suitable admissibility condition, we prove that the polynomials f&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=b98d65784e9fe2f8fa077aa1517be738">f" data-inlimgeid="1-s2.0-S0021904514000938-si9.gif">flow="scroll">mna,c;F, formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000938&_mathId=si10.gif&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=ff8badf24791bbba2d2b24af47b1791b" title="Click to view the MathML source">n∈蟽_Fflow="scroll">n∈蟽F, are actually exceptional Meixner polynomials; that is, in addition, they are orthogonal and complete with respect to a positive measure. By passing to the limit, we transform the Casorati determinant of Meixner polynomials into a Wronskian type determinant of Laguerre polynomials f&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=b1cfeec33977b13b1825d6577a4d297b">f" data-inlimgeid="1-s2.0-S0021904514000938-si11.gif">flow="scroll">(Ln伪)n. Under the admissibility conditions for formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000938&_mathId=si12.gif&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=07da2f6d8821666ca40e2ef811d2cb3f" title="Click to view the MathML source">Fflow="scroll">F and formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904514000938&_mathId=si13.gif&_user=111111111&_pii=S0021904514000938&_rdoc=1&_issn=00219045&md5=51a789cd446a13621cf465f1b096d221" title="Click to view the MathML source">伪flow="scroll">伪, these Wronskian type determinants turn out to be exceptional Laguerre polynomials.