New properties of the Lerch's transcendent

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• 作者：E.M. Ferreira^a ; ^{erasmo@if.ufrj.br" class="auth_mail" title="E-mail the corresponding author} ; A.K. Kohara^a ; ^{kendi@if.ufrj.br" class="auth_mail" title="E-mail the corresponding author} ; J. Sesma^b ; ^{javier@unizar.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11M35 ; 30E15
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：172
• 期：Complete
• 页码：21-31
• 全文大小：290 K

文摘

A new representation of the Lerch's transcendent class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302219&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302219&_rdoc=1&_issn=0022314X&md5=a479cb863b5946919242b66d5d93fdee" title="Click to view the MathML source">Φ(z,s,a)class="mathContainer hidden">class="mathCode">$\mathrm{\Phi }(z,s,a)$, valid for positive integer class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302219&_mathId=si2.gif&_user=111111111&_pii=S0022314X16302219&_rdoc=1&_issn=0022314X&md5=1f8d7876296e8a1967ae262a78d93c64" title="Click to view the MathML source">s=n=1,2,…class="mathContainer hidden">class="mathCode">$s=n=1,2,\dots$ and for z and a   belonging to certain regions of the complex plane, is presented. It allows to write an equation relating class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302219&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302219&_rdoc=1&_issn=0022314X&md5=da03e9ee43c07e52bbfe9d1e81858757" title="Click to view the MathML source">Φ(z,n,a)class="mathContainer hidden">class="mathCode">$\mathrm{\Phi }(z,n,a)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302219&_mathId=si4.gif&_user=111111111&_pii=S0022314X16302219&_rdoc=1&_issn=0022314X&md5=5af4746279fae484fdf2bcaf7360029d" title="Click to view the MathML source">Φ(1/z,n,1−a)class="mathContainer hidden">class="mathCode">$\mathrm{\Phi }(1/z,n,1-a)$, which in turn provides an expansion of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302219&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302219&_rdoc=1&_issn=0022314X&md5=da03e9ee43c07e52bbfe9d1e81858757" title="Click to view the MathML source">Φ(z,n,a)class="mathContainer hidden">class="mathCode">$\mathrm{\Phi }(z,n,a)$ as a power series of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302219&_mathId=si5.gif&_user=111111111&_pii=S0022314X16302219&_rdoc=1&_issn=0022314X&md5=ec4ce4d133a9f8a2cbbbfae0dc616aca" title="Click to view the MathML source">1/zclass="mathContainer hidden">class="mathCode">$1/z$, convergent for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302219&_mathId=si6.gif&_user=111111111&_pii=S0022314X16302219&_rdoc=1&_issn=0022314X&md5=4446d0a5e1d4f5a7d4551527b893855c" title="Click to view the MathML source">|z|>1class="mathContainer hidden">class="mathCode">$|z|>1$.