Wikipedia :
Polynomial
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Calculus with polynomials
Sheffer sequence (Poweroid)
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Stone-Weierstrass theorem
cos(nq) is a polynomial function of cos(q). The following relation defines a polynomial of degree n known as the Chebyshev polynomial of degree n:
cos (nq) = Tn (cos q)
The symbol T comes from careful Russian transliterations like Tchebyshev, Tchebychef (French) or Tschebyschow (German). Alternate spellings include Tchebychev (French) and "Chebychev".
The trigonometric formula cos (n+2)q = 2 cos q cos (n+1)q - cos nq translates into a simple recurrence relation which makes Chebyshev polynomials very easy to tabulate, namely:
Knowing only the highest term of Tn and its obvious n distinct real zeroes, we obtain immediately Tn as a product of n factors:
The case Tn (0) = (-1)n tells something nice about a product of cosines.
Chebyshev polynomials verify Tm(Tn(x)) = Tmn(x). This unique property makes it possible to define pairs of closely related functions from any pair of arithmetic functions u and v (with subexponential growth) that are Dirichlet inverses of each other, using the following symmetrical relations:
If f (0) = 0, those series are usually absolutely convergent, because Tn(x) decreases exponentially with n, for any fixed x in ]-1,+1[.
Proof : Expand the latter right-hand-side using the definition of g :
å m å n u(n) v(m) f ( Tmn (x) ) = å k [ å d|k u(d) v(k/d) ] f ( Tk (x) )
u and v being Dirichlet inverses, the bracket is either 1 (if k = 1) or 0. QED
This applies, in particular, when u is a totally multiplicative arithmetic function [i.e., such that u(mn) = u(m) u(n) for any m & n ] in which case its Dirichlet inverse can be expressed using the Möbius function (m) :
v(n) = m(n) u(n)
Using Tn(x) = x1/n instead of Chebyshev polynomials, this pattern was used in 1859 by Riemann to link his (normalized) prime-counting function f = p with the celebrated jump function g = J he obtained with u(n) = 1/n.
Bienaymé-Chebyshev inequality | Chebyshev economization | Pafnuty Chebyshev (1821-1894)
They obey exactly the same second-order recurrence relation as the above Chebychev polynomials of the first kind but the starting points are different:
Generalized
Chebychev Polynomials, Planar Trees and Galois Theory by Anton Bankevich (2008年02月28日)
Wikipedia :
Chebyshev polynomials
of the first and second kinds.
Dessins d'enfants, trees and Shabat polynomials.
The Legendre polynomials (A008316) are recursively defined by:
They are linked to the expressions of spherical harmonics in terms of the colatitude q Î [0,p[ and the longitude f (modulo 2p).
Electric multipoles
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Figure of the Earth
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Dynamic form factors
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Legendre Polynomials
Adrien-Marie Legendre (1752-1833)
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MathWorld :
Legendre Polynomials
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Spherical Harmonics
Laguerre's equation is a second-order linear differential equation:
x y'' + (1-x) y' + n y = 0
It has non-singular solutions only when n is a non-negative integer. In that case, a solution is Ln(n), the Laguerre polynomial of order n given by:
Sorin is credited for the following generalized Laguerre equation :
x y'' + (a+1-x) y' + n y = 0
This is satisfied by the Laguerre function, defined by:
Because of the way binomial coefficients vanish, a polynomial (a finite sum) called associated Laguerre polynomial is so obtained when n is a non-negative integer. Otherwise, the above is a divergent series which is Borel-summable.
Ordinary Laguerre polynomials correspond to the special case a = 0.
Wikipedia :
Associated Laguerre Polynomials
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Nikolay Sonin (1849-1915)
Rook Polynomials
of John Riordan (1903-1988)
"On Laguerre's Series"
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First note
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second note
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third note
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by Einar Hille (1926).
MathWorld :
Laguerre Polynomials
The above are more popular than the simpler modified Hermite polynomials Hen which can be defined via: Hn (x) = 2n/2 Hen (2½ x)
Hermite Polynomials (Wikipedia)
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Hermite Polynomials (MathWorld)
Charles Hermite (1822-1901; X1942)
The reverse Bessel polynomials tabulated below appear in the transfer functions of Bessel-Thomson filters
MathWorld : Bessel polynomials
Like Hermite polynomials and Euler polynomials, the sequence of Bernoulli polynomials start with some nonzero constant polynomial (namely, 1) and subsequently verify the Appell property, which is to say:
dBn (x) / dx = n Bn-1 (x)
This relation becomes a recursive definition if the successive constants of integration are given as a prescribed sequence:
Bn = Bn (0)
The polynomials can be expressed in terms of that sequence of numbers:
With the convention adopted in Numericana (i.e., B1 = ½) we have:
Authors who posit that B1 = -½ specify instead that Bn = Bn(0). Note that those two conventions only differ in the case n = 1.
The Bernoulli polynomials are not affected by the choice of convention for Bernoulli numbers. Neither are relations between those polynomials, like:
Bn (1 - x) = (-1)n Bn (x)
Bn (1 + x) = Bn (x) + n xn-1
Besides the aforementioned case n = 1, Bn vanishes for odd values of n.
By the von Staudt-Clausen theorem (1840) the denominator of B2n is the product of all primes p for which p-1 divides 2n.
Introduction to Bernoulli and
Euler Polynomials by Zhi-Wei Sun,
Nanjing University (2002年06月06日)
Mathworld
}
Wikipedia
}
Encyclopedia of Mathematics
}
Kim Milton
von Staudt-Clausen theorem (1840)
}
Karl von Staudt (1798-1867)
}
Thomas Clausen (1801-1885)
Darboux's summation formula
}
Umbral calculus
Why do Bernoulli numbers
arise everywhere? (MathOverflow, since 2011).
After deriving explicit formulas up to p = 17, Johann Faulhaber observed that, if p = 2q+1 is odd, then the sum of the p-th powers of the integers from 0 to n is a polynomial of degree q+1 in the variable x = n(n+1)/2. A related expression holds for a nonzero even p, namely:
That result was proved in full generality by Carl Jacobi, in 1834.
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Faulhaber's formula
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Faulhaber polynomials
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Johann Faulhaber of Ulm (1580-1635)
Johann
Faulhaber and Sums of Powers by Donald E. Knuth
(Math. Comp, 61, 203, 277-294, July 1993).
En(x)
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Euler numbers can be expressed in terms of the above Euler polynomials:
En = 2n En (½)
The Euler numbers of odd index vanish. The signs of even-indexed Euler numbers alternate.
Leonhard Euler (1707-1783)
MathWorld :
Euler polynomials
Mittag-Leffler polynomials were first discussed under that name in 1940, by Harry Bateman (1882-1946).
They obey the same binomial formula as ordinary powers:
Mittag-Leffler Polynomials (1891)
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MathWorld
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Gösta Mittag-Leffler (1846-1927)
Umbral calculus
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Sheffer sequence (poweroids)
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Isador M. Sheffer (1901-1992,
PhD 1927)
The polynomial of Mittag-Leffler by
Harry Bateman (1882-1946) Proc. N.A.S., 26, 491-496 (1940年07月13日).
Generalization of power polynomials
by John D. Cook (2020年01月28日).
[画像: Come back later, we're still working on this one... ]
Umbral calculus
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Sheffer sequence (poweroids)
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Isador M. Sheffer (1901-1992,
PhD 1927)
Polynomials of binomial type
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Cumulants
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Moments
The nth cyclotomic polynomial Fn is the unique monic polynomial dividing x k - 1 for k = n but not for any lesser value of k.
When n > 1, Fn is palindromic. If n has at most two distinct odd prime factors, then the coefficients of Fn stay within {-1,0,1}. That holds for n < 105; the first product of three distinct odd primes (Adolph Migotti, 1883). Those coefficients can be arbitrarily large (Issai Schur, 1931). Furthermore, any given integer occurs as a coefficient of some cyclotomic polynomial (Jiro Suzuki, 1987).
Fn is an irreducible polynomial over the rationals, whose degree is equal to the Euler totient f (n). That nontrivial fact is due to Carl F. Gauss.
The following definition also holds for n = 0 (as an empty product is 1).
For n > 0, the cyclotomic polynomial Fn can thus be defined as the unique monic polynomial whose roots are the primitive nth roots of unity.
As with any multiplicative function, the (rarely used) value of f at zero is f (0) = 0 (as its one-line definition implies) which confirms F0 = 1.
Unfortunately, the OEIS (A013595) is still following a dubious ad-hoc definition for F0 from Maple® (albeit with due apologies).
The following factorization yields as many factors as there are divisors of n:
Equating polynomial degrees retrieves a Dirichlet convolution: N = u*f
The following interesting equation involves the Möbius function m :
In this, all the negative exponents do cancel "miraculously".
Encyclopedia of Mathematics
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MathWorld
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Wikipedia
"Some properties of coefficients of cyclotomic polynomials"
Marcin Mazur,
Bogdan V. Petrenko (2019年02月12日).
"Cyclotomic polynomials"
by Brett Porter (student project at Whitman College, 2015年05月20日).
Bungers-Lehmer Theorem on Cyclotomic
Coefficients by Robin Whitty
(Theorem of the Day #175).
Conditional proof in the 1934 thesis of Rolf Bungers,
possibly the future seismologist (1909-1942)
[former student of Gustav Angenheister (1878-1945)
who died in a plane crash in Norway, on 1942年12月24日]
On the magnitude of the coefficients of the cyclotomic polynomial
(June 1936) Emma Lehmer (1906-2007)
Cyclotomic polynomials (19:42)
by MathDoctorBob (2013年01月11日).
Tips and tricks for computing cyclotomic polynumbers (27:09)
by Norman J. Wildberger (2020年09月12日).
A polynomial Pp can be defined for which the following identity holds, which provides a nontrivial factorization of some special integers:
( p x2 ) p - (-1)m = ( p x2 - (-1)m ) Pp (-x) Pp (x)
Here, p = 2m+1 is an odd prime
(see Sophie Germain identity for p=2).
Pp (x) = Ap ( p x2 ) + (p x) Bp ( p x2 )
where Ap and Bp are both
palindromic
monic polynomials.
Ap has degree m. Bp has degree m-1.
For p=31 (and x=9) this factors a nice 102-digit
semiprime: (251131+1) / 2512
=わ 889923919072997985238634558820908333948499157179463
× 1111413273683146858652465162019244587926917356315577
That factorization would take a long time with a general-purpose program.
For compactness, we'll give palindromic polynomials as lists of coefficients with underlined central ones (so the mirror endings can be freely truncated).
A37- =
(1, 19, 79, 183, 285, 349, 397, 477, 579, 627, 579, 477, 397, 349, 285, 183, 79, 19, 1)
B37- =
(1, 7, 21, 39, 53, 61, 71, 87, 101, 101, 87, 71, 61, 53, 39, 21, 7, 1)
A41- =
(1, 21, 67, 49, 7, 35, 15, 11, -23, -65, -31, -65, -23, 11, 15, 35, 7, 49, 67, 21, 1)
B41- =
(1, 7, 11, 3, 3, 5, 1, 1, -9, -7, -7, -9, 1, 1, 5, 3, 3, 11, 7, 1)
A43+ =
(1, 21, 81, 169, 223, 225, 213, 223, 229, 197, 159,
159, 197, 229, 223, 213, 225, 223, 169...
B43+ =
(1, 7, 19, 31, 35, 33, 33, 35, 33, 27, 23, 27, 33, 35, 33, 33, 35, 31, 19, 7, 1)
A47+ = (1, 23, 65, -15, -169, -97, 179, 287, -37, -375, -149, 311,
311, -149, -375, -37, 287, 179...
B47+ = (1, 7, 7, -15, -25, 5, 41, 25, -37, -49, 15, 57, 15, -49, -37, 25, 41, 5, -25, -15, 7, 7, 1)
A53- = (1, 27, 113, 103, -155, -219, 263, 513, -59, -465, 75, 551, 93, -357,
93, 551, 75, -465, -59...
B53- = (1, 9, 19, -1, -35, -3, 67, 41, -51, -39, 57, 57, -31,
-31, 57, 57, -39, -51, 41, 67, -3, -35, -1...
A59+ = (1, 29, 111, 55, -85, 47, 11, 53, 131, -245, 41, 103, -111, 227, -103,
-103, 227, -111, 103...
B59+ = (1, 9, 15, -5, -5, 9, -3, 21, -9, -25, 25, -11, 9, 19, -31,
19, 9, -11, 25, -25, -9, 21, -3, 9, -5, -5...
A61- = (1,31,191,637,1541,2979,4881,7029,9125,10953,12397,13511,14379,
15053,15511,15667...
B61- = (1, 11, 47, 131, 281, 497, 761, 1037, 1291, 1501, 1663, 1789, 1887, 1961,
2001, 2001...
A67+ = (1,33,193,565,1055,1429,1599,1803,2225,2637,2617,2195,1869,1875,1865,1469,991,991...
B67+ = (1, 43, 99, 155, 187, 205, 243, 301, 329, 297, 243, 225, 233, 209, 147, 111, 147, 209, 233...
A71+ = (1, 35, 169, 155, -109, 233, 597, 39, 101, 445, 163, 293, 89, -203, 249, -49, -505, 37,
37...
B71+ = (1, 11, 25, 1, -5, 63, 43, -9, 43, 37, 21, 35, -19, 1, 29, -47, -35, 23,
-35, -47, 29, 1, -19, 35...
For p=61 (with x=2) this gives the
factorization
of the 144-digit semiprime (24461-1) / 35 =
254180335737792836487420059360430288526895310810588085366845580859576779
× 691880648894768106905652479597579967344338476040716833288367161850591919
(p x2)p ± 1 = ( p x2 ± 1) [ Ap± (p x2)2 - p2 x2 Bp± (p x2)2 ]
As a polynomial identity, that's equivalent (using y = p x2 ) to a simpler relation (albeit not directly applicable to integer factorization):
yp ± 1 = (y ± 1) [ Ap± (y)2 - (p y) Bp± (y)2 ] (with ±1 = (-1)(p+1)/2 )
"Formules de Cauchy et de Lejeune-Dirichlet"
by Edouard Lucas
(Compte Rendu,
pp. 164-173. 1878年08月29日)
Prime Numbers and Computer Methods for Factorization
(Table 24, p. 444) by Hans Riesel (1929-2014).
The Cunningham project
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Numericana :
Aurifeuillian Factorizations and Beyond
This can be used to factor x5 + x4 + 1 :
My first quintic equation (10:28) by Steve Chow (blackpenredpen, 2020年06月03日).
