Multivariate delta Gončarov and Abel polynomials

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• 作者：Rudolph Lorentz^a ; ^{rudolph.lorentz@qatar.tamu.edu" class="auth_mail" title="E-mail the corresponding author} ; Salvatore Tringali^a ; ¹ ; ^{salvatore.tringali@uni-graz.at" class="auth_mail" title="E-mail the corresponding author} ; Catherine H. Yan^b ; ^{cyan@math.tamu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Abel and Gončarov polynomials ; Appell relations ; Delta operators ; Interpolation ; Umbral calculus
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：663-680
• 全文大小：453 K

文摘

Classical Gončarov polynomials are polynomials which interpolate derivatives. Delta Gončarov polynomials are polynomials which interpolate delta operators, e.g., forward and backward difference operators. We extend fundamental aspects of the theory of classical bivariate Gončarov polynomials and univariate delta Gončarov polynomials to the multivariate setting using umbral calculus. After introducing systems of delta operators, we define multivariate delta Gončarov polynomials, show that the associated interpolation problem is always solvable, and derive a generating function (an Appell relation) for them. We show that systems of delta Gončarov polynomials on an interpolation grid class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305042&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305042&_rdoc=1&_issn=0022247X&md5=b85cd1bae1a4ea776448545fd0710ca6" title="Click to view the MathML source">Z⊆R^dclass="mathContainer hidden">class="mathCode">$Z\subseteq {\mathbb{R}}^d$ are of binomial type if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305042&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305042&_rdoc=1&_issn=0022247X&md5=4be57bbb8b07f3b3ef6746b38cd18ed9" title="Click to view the MathML source">Z=AN^dclass="mathContainer hidden">class="mathCode">$Z=A{\mathbb{N}}^d$ for some class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305042&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305042&_rdoc=1&_issn=0022247X&md5=7937145cad583cbb0a8e693987f64a21" title="Click to view the MathML source">d×dclass="mathContainer hidden">class="mathCode">$d\times d$ matrix A. This motivates our definition of delta Abel polynomials to be exactly those delta Gončarov polynomials which are based on such a grid. Finally, compact formulas for delta Abel polynomials in all dimensions are given for separable systems of delta operators. This recovers a former result for classical bivariate Abel polynomials and extends previous partial results for classical trivariate Abel polynomials to all dimensions.