Dixon multinomial expression

It is L. with Dixon multinomial expression (I will burn Dixon and come British: Dickson polynomials) or the ブリューワ multinomial expression (Brewer polynomials) in mathematics E. I am introduced by Dickson (1897) and, in a certain multinomial expression line rediscovered in the study of the ブリューワ sum by Brewer (1961), am described with D_n(x,α).

It is the equivalent essentially, and, actually, on the complex number body, the Dixon multinomial expression is often called a Chebyshev polynominal by the Dixon multinomial expression with a Chebyshev polynominal by variable conversion. When a Dixon multinomial expression is not a Chebyshev polynominal and the equivalent, it is studied a lot on a limited body. One of the interest includes that the Dixon multinomial expression gives many examples of the substituted multinomial expression (English version) for fixed α. But the substituted multinomial expression is a multinomial expression to act as substitution of the limited body.

Table of contents

Definition

Is D₀(x,α) = 2; n > For 0 the Dixon multinomial expression is given in next (first-class).

${\displaystyle D_{n}(x,\alpha) =\sum _{p=0}^{\lfloor n/2\rfloor }{\frac {n}{n-p}}{\binom {n-p}{p}}(-\alpha) ^{p}x^{n-2p}.}$

I am as follows when I give some of this beginning.

${\displaystyle D_{0}(x,\alpha) =2\,}$

${\displaystyle D_{1}(x,\alpha) =x\,}$

${\displaystyle D_{2}(x,\alpha) =x^{2}-2\alpha \,}$

${\displaystyle D_{3}(x,\alpha) =x^{3}-3x\alpha \,}$

${\displaystyle D_{4}(x,\alpha) =x^{4}-4x^{2}\alpha +2\alpha ^{2}.\,}$

Second kind Dixon multinomial expression E_n is defined in next.

${\displaystyle E_{n}(x,\alpha) =\sum _{p=0}^{\lfloor n/2\rfloor }{\binom {n-p}{p}}(-\alpha) ^{p}x^{n-2p}.}$

A lot of these studies are not freed, and the property is similar to a first-class Dixon multinomial expression. I am as follows when I give some of the second kind Dixon multinomial expression of beginning.

${\displaystyle E_{0}(x,\alpha) =1\,}$

${\displaystyle E_{1}(x,\alpha) =x\,}$

${\displaystyle E_{2}(x,\alpha) =x^{2}-\alpha \,}$

${\displaystyle E_{3}(x,\alpha) =x^{3}-2x\alpha \,}$

${\displaystyle E_{4}(x,\alpha) =x^{4}-3x^{2}\alpha +\alpha ^{2}.\,}$

Property

D_n is the next equation

${\displaystyle D_{n}(u+\alpha /u,\alpha) =u^{n}+(\alpha /u)^{n}\,;}$

${\displaystyle D_{mn}(x,\alpha) =D_{m}(D_{n}(x,\alpha) ,\alpha ^{n})\,}$

I satisfy を. For n≥2, the Dixon multinomial expression is 漸化式

${\displaystyle D_{n}(x,\alpha) =xD_{n-1}(x,\alpha) -\alpha D_{n-2}(x,\alpha) \,}$

${\displaystyle E_{n}(x,\alpha) =xE_{n-1}(x,\alpha) -\alpha E_{n-2}(x,\alpha) \,}$

I satisfy を. Dixon multinomial expression Dn = y is a solution of the next differential equation that it is the way it goes.

${\displaystyle (x^{2}-4\alpha) y''+xy'-n^{2}y=0.\,}$

In addition, second kind Dixon multinomial expression En = y is a solution of the next differential equation.

${\displaystyle (x^{2}-4\alpha) y''+3xy'-n(n+2)y=0.\,}$

Those conventional generating function is given in next.

${\displaystyle \sum _{n}D_{n}(x,\alpha) z^{n}={\frac {2-xz}{1-xz+\alpha z^{2}}}\,}$

${\displaystyle \sum _{n}E_{n}(x,\alpha) z^{n}={\frac {1}{1-xz+\alpha z^{2}}}.\,}$

with other multinomial expressions of relationships

• The Dixon multinomial expression in the complex number health is connected with Chebyshev polynominal T_n and U_n in the next expression.

${\displaystyle D_{n}(2xa,a^{2})=2a^{n}T_{n}(x)\,}$

${\displaystyle E_{n}(2xa,a^{2})=a^{n}U_{n}(x).\,}$

It is an important thing, but Dixon multinomial expression D_n(x,a) can define ring and 標数 where a is not square on 2 rings. In such a case D_n(x,a) will not often have connection with the Chebyshev polynominal.

• The Dixon multinomial expression that a parameter is α = 1 or α = -1 is connected with Fibonacci multinomial expression and リュカ multinomial expression.
• The Dixon multinomial expression in case of α = 0 gives the next monomial expression:

${\displaystyle D_{n}(x,0)=x^{n}\,.}$

Substituted multinomial expression and Dixon multinomial expression

I call a thing working as original substitution of the body the substituted multinomial expression (permutation polynomial) (for the given limited body).

The necessary and sufficient condition for Dixon multinomial expression D_n(x,α) (considered to be the function of x for fixed α) to be the substituted line for the body having the material of the q unit n and q²-1 each other bare であることである [¹].

M. Fried (1970) showed that the arbitrary integer multinomial expression that was the substituted line for many bare bodies was composition of a Dixon multinomial expression and (the Yuri coefficient) linear multinomial expression endlessly. This claim was known as expectation of シューア, but シューア did not really perform the expectation. Because the article of Fried included many mistakes; the correction is G. Do it by Turnwald (1995); P. Müller (1997) gave the concise proof along the discussion with シューア.

It is P. more A degree showed that any substitution multinomial expression on q-1 and limited body F_q which was smaller each other than bare で and q^1/4 was composition of a Dixon multinomial expression and the linear shape multinomial expression by all means to Müller (1997).

References

1. ^ Lidl & Niederreiter (1997) p.356

• Brewer, B. W. (1961), "On certain character sums," it is 241–245, doi: Transactions of the American Mathematical Society 99 10.2307/1993392, ISSN 0002-9947, MR0120202, Zbl 0103.03205, http://www.jstor.org/stable/1993392 
• Dickson, L.E. (1897). "The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group I,II." Ann. Of Math. (The Annals of Mathematics) 11 (1/6): 65–120; 161–183. doi: 10.2307/1967217. ISSN 0003-486X. JFM 28.0135.03. JSTOR 1967217. 
• Fried, Michael (1970). "On a conjecture of Schur". Michigan Math. J. 17: 41–55. doi: 10.1307/mmj/1029000374. ISSN 0026-2285. MR0257033. Zbl 0169.37702. http://projecteuclid.org/euclid.mmj/1029000374. 
• Lidl, R.; Mullen, G. L.; Turnwald, G. (1993). Dickson polynomials. Pitman Monographs and Surveys in Pure and Applied Mathematics. 65. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York. ISBN 0-582-09119-5. MR1237403. Zbl 0823.11070. 
• Lidl, Rudolf; Niederreiter, Harald (1997). Finite fields. Encyclopedia of Mathematics and Its Applications. 20 (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. Zbl 0866.11069. 
• Mullen, Gary L. (2001), "Dixon multinomial expression," it is in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1,556,080,104, http://eom.springer.de/D/d120140.htm 
• Müller, Peter (1997). "A Weil-bound free proof of Schur's conjecture". Finite Fields Appl. 3: 25–32. doi: 10.1006/ffta.1996.0170. It is http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WFM-45M8VYC-J&_user=99318&_coverDate=01%2F31%2F1997&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000007678&_version=1&_urlVersion=0&_userid=99318&md5=b40a5d38475bafcd5b0df1826f48680f&searchtype=a. Zbl 0904.11040 
• Rassias, Thermistocles M.; Srivastava, H.M.; Yanushauskas, A. (1991). Topics in Polynomials of One and Several Variables and Their Applications: A Legacy of P.L.Chebyshev. World Scientific. pp. 371–395. ISBN 981-02-0614-3. 
• Turnwald, Gerhard (1995). "On Schur's conjecture". J. Austral. Math. Soc. Ser. A 58 (03): 312–357. doi: 10.1017/S1446788700038349. MR1329867. It is http://journals.cambridge.org./action/displayFulltext?type=1&pdftype=1&fid=4986396&jid=JAZ&volumeId=58&issueId=&aid=4986388. Zbl 0834.11052 
• Young, Paul Thomas (2002). "On modified Dickson polynomials". Fib. Quaterly 40 (1): 33–40. http://www.fq.math.ca/Scanned/40-1/young.pdf. 

It is acquired by "https://ja.wikipedia.org/w/index.php?title= Dixon multinomial expression &oldid=51926674"

This article is taken from the Japanese Wikipedia Dixon multinomial expression

This article is distributed by cc-by-sa or GFDL license in accordance with the provisions of Wikipedia.

Wikipedia and Tranpedia does not guarantee the accuracy of this document. See our disclaimer for more information.

In addition, Tranpedia is simply not responsible for any show is only by translating the writings of foreign licenses that are compatible with CC-BY-SA license information.