|
The identity is quite useful in conjunction with the recursive generating formula, inasmuch as it enables one to calculate the cosine of any integer multiple of an [[angle]] solely in terms of the cosine of the base angle.
The first two Chebyshev polynomials of the first kind arecan be computed directly from the definition[[Euler's to beidentity]]
:<math>\cos k \theta + i \sin k \theta = e^{k \theta} = (e^{i \theta})^k = (\cos \theta + i \sin \theta)^k.</math>
Expanding the latter, one gets
:<math>(\cos \theta + i \sin \theta)^k = \sum\limits_{j=0}^k i^j \sin^j \theta \cos^{k-j} \theta.</math>
Then, to get the expression for <math>\cos k \theta</math>, one should look on the real part of the expression, which is obtained from summands corresponding to even indices. Noting <math>i^{2j} = (-1)^j</math> and <math>\sin^{2j} \theta = (1-\cos^2 \theta)^j</math>, one gets the explicit formula
:<math>\cos k \theta = \sum\limits_{j=0}^{\lfloor k / 2 \rfloor} \binom{k}{2j} (\cos^2 \theta - 1)^j \cos^{k-j} \theta,</math>
which in turn means that
:<math>T_k(x) = \sum\limits_{j=0}^{\lfloor k / 2 \rfloor} \binom{k}{2j} (x^2-1)^j x^{k-2j}.</math>
Alternatively, the first two Chebyshev polynomials of the first kind are computed directly from the definition to be
:<math>T_0(\cos\theta) = \cos(0\theta) = 1</math>
and
{\tilde V}_n(2x,1) &= 2\, T_n(x).
\end{align}</math>
It follows that they also satisfy a pair of mutual recurrence equations:<ref name="BatemanVol2">{{cite book |editorauthor-lastlast1=Erdélyi |editorauthor-firstfirst1=Arthur |editor-linkurl=Arthur Erdélyi http://apps.nrbook.com/bateman/Vol2.pdf<!-- Director:print run "7 8 9 10 11 12 13 HL 9 8 7 6" 1981 with 1 errata page (14 entries) -->|author-last1=Erdélyi |author-first1title=ArthurHigher |author-link1=ArthurTranscendental ErdélyiFunctions <!-- ResearchVolume associates:II --> Based, in part, on notes left by Harry Bateman. |author-last2=Magnus |author-first2=(Hans Heinrich) Wilhelm |author-link2=Hans Heinrich Wilhelm Magnus |author-last3=Oberhettinger |author-first3=Fritz |author-link3=:de:Fritz Oberhettinger |author-last4=Tricomi |author-first4=Francesco Giacomo |author-link4last5=FrancescoBertin Giacomo Tricomi <!-- Research assistants: -->|author-first5=David |author-last5last6=BertinFulks |author-first6=Watson B. |author-last6last7=FulksHarvey |author-first7=Albert Raymond |author-last7last8=HarveyThomsen, Jr. |author-first8=Donald L. |author-last8last9=Thomsen, Jr.Weber |author-first9=Maria A. |author-last9date=Weber1953 |author-first10publisher=Eoin[[McGraw-Hill LairdBook Company, Inc.]] |authoreditor-last10last=WhitneyErdélyi |authoreditor-link10first=:d:Q102128557Arthur |editor-link=Arthur Erdélyi <!-- Vari-typistDirector: -->|author-first11=Rosemarie |author-last11edition=Stampfel |title=Higher Transcendental Functions - Volume II - Based, in part, on notes left by Harry Bateman.1 |series=Bateman Manuscript Project |volume=II |publisher=[[McGraw-Hill Book Company, Inc.]] |publication-place=New York / Toronto / London |datepage=1953 |edition=1184:(3),(4) |lccn=53-5555 |id=Contract No. N6onr-244 Task Order XIV. Project Designation Number: NR 043-045. Order No. 19546 |urlauthor-link1=http://apps.nrbook.com/bateman/Vol2.pdfArthur Erdélyi <!-- printResearch runassociates: "7--> 8|author-link2=Hans 9Heinrich 10Wilhelm 11Magnus 12|author-link3=:de:Fritz 13Oberhettinger HL|author-link4=Francesco 9Giacomo 8Tricomi 7<!-- 6"Research 1981 with 1 errata page (14 entries)assistants: --> |access-date=2020-07-23 |url-status=live |archive-url=https://web.archive.org/web/20170409113446/http://apps.nrbook.com/bateman/Vol2.pdf |archive-date=2017-04-09 |url-status=live |author-first10=Eoin Laird |author-last10=Whitney |author-link10=:d:Q102128557 <!-- Vari-typist: --> |author-first11=Rosemarie |author-last11=Stampfel}} [https://authors.library.caltech.edu/43491/7/Volume%202.pdf][https://resolver.caltech.edu/CaltechAUTHORS:20140123-104529738]<!-- https://authors.library.caltech.edu/43491/ --> (xvii+1 errata page<!-- (14 entries) in print run "III 19546" -->+396 pages, red cloth hardcover) (NB. Copyright was renewed by [[California Institute of Technology]] in 1981.); Reprint: Robert E. Krieger Publishing Co., Inc., Melbourne, Florida, USA. 1981. {{ISBN|0-89874-069-X}}; Planned Dover reprint: {{ISBN|0-486-44615-8}}.
|page=184:(3),(4)}} [https://authors.library.caltech.edu/43491/7/Volume%202.pdf<!-- print run "IV 19546" without errata pages --><!-- https://web.archive.org/web/20200704144437/https://authors.library.caltech.edu/43491/7/Volume%202.pdf -->][https://resolver.caltech.edu/CaltechAUTHORS:20140123-104529738<!-- https://web.archive.org/web/20190119121233/https://authors.library.caltech.edu/43491/ -->]<!-- https://authors.library.caltech.edu/43491/ --> (xvii+1 errata page<!-- (14 entries) in print run "III 19546" -->+396 pages, red cloth hardcover) (NB. Copyright was renewed by [[California Institute of Technology]] in 1981.); Reprint: Robert E. Krieger Publishing Co., Inc., Melbourne, Florida, USA. 1981. {{ISBN|0-89874-069-X}}; Planned Dover reprint: {{ISBN|0-486-44615-8}}.
</ref>
:<math>\begin{align}
| 