The prompter method: A treatment for hard-to-solve iterative functional equations

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• 作者：Hideaki Izumi ^{izumi.hideaki@it-chiba.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Iterative functional equation ; Formal solution ; Iterative root
• 刊名：Journal of Symbolic Computation
• 出版年：2016
• 出版时间：July-August 2016
• 年：2016
• 卷：75
• 期：Complete
• 页码：193-208
• 全文大小：370 K

文摘

The author developed a new method for obtaining formal series solutions to polynomial-like iterative functional equations of the form 5001133&_mathId=si1.gif&_user=111111111&_pii=S0747717115001133&_rdoc=1&_issn=07477171&md5=8ec3b9aa2e66d354e61fe55d491aa9f5">5001133-si1.gif">$\sum _{n=1}^N{a}_n{f}^n(x)=g(x)$, where 5001133&_mathId=si2.gif&_user=111111111&_pii=S0747717115001133&_rdoc=1&_issn=07477171&md5=7ef9b7b10d167283dc975183962a6d9f">5001133-si2.gif"> is the n-th iterate of an unknown function f   and where 5001133&_mathId=si3.gif&_user=111111111&_pii=S0747717115001133&_rdoc=1&_issn=07477171&md5=1d08609b70b907f66021dc3d4dbeb477" title="Click to view the MathML source">g(x)$g(x)$ is a promptered exponential series, namely, the sum of a Dirichlet series and a linear term called prompter. In this method, a formal composition 5001133&_mathId=si4.gif&_user=111111111&_pii=S0747717115001133&_rdoc=1&_issn=07477171&md5=52c7c4b2d31e219d884257e55cf64428" title="Click to view the MathML source">f₁∘f₂$f_1\circ f_2$ of two promptered exponential series 5001133&_mathId=si5.gif&_user=111111111&_pii=S0747717115001133&_rdoc=1&_issn=07477171&md5=8ddfe19cb3cc58c7548c9f56264302fa" title="Click to view the MathML source">f₁$f_1$ and 5001133&_mathId=si6.gif&_user=111111111&_pii=S0747717115001133&_rdoc=1&_issn=07477171&md5=0c3031b5324a24dde87b5f88f10a5eb5" title="Click to view the MathML source">f₂$f_2$, where the coefficient of the prompter of 5001133&_mathId=si6.gif&_user=111111111&_pii=S0747717115001133&_rdoc=1&_issn=07477171&md5=0c3031b5324a24dde87b5f88f10a5eb5" title="Click to view the MathML source">f₂$f_2$ is positive, plays a crucial rôle. We also solve the equation above where 5001133&_mathId=si3.gif&_user=111111111&_pii=S0747717115001133&_rdoc=1&_issn=07477171&md5=1d08609b70b907f66021dc3d4dbeb477" title="Click to view the MathML source">g(x)$g(x)$ is a promptered trigonometric series.