formula. The partial or incomplete exponential Bell polynomials are a triangular array of polynomials given by B n , k ( x 1 , x 2 , … , x n − k + 1 )...

Touchard polynomials, studied by Jacques Touchard (1939), also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence...

for convergence. He is the eponym of the Bell polynomials and the Bell numbers of combinatorics. In 1924 Bell was awarded the Bôcher Memorial Prize for...

All-one polynomials Abel polynomials Bell polynomials Bernoulli polynomials Cyclotomic polynomials Dickson polynomials Fibonacci polynomials Lagrange...

notebook, he investigated both Bell polynomials and Bell numbers. Early references for the Bell triangle, which has the Bell numbers on both of its sides...

to define the multidimensional polynomials. Like the other classical orthogonal polynomials, the Hermite polynomials can be defined from several different...

of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable...

All one polynomials Appell sequence Askey–Wilson polynomials Bell polynomials Bernoulli polynomials Bernstein polynomial Bessel polynomials Binomial...

Gram–Charlier A series. Such an expansion can be written compactly in terms of Bell polynomials as exp ⁡ [ ∑ r = 3 ∞ κ r ( − D ) r r ! ] = ∑ n = 0 ∞ B n ( 0 , 0 ,...

of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type). Polynomial sequences...

mathematics Bell polynomials, in mathematics Bell state, in quantum information science Diving bell, a cable-suspended underwater airtight chamber Bell station...

the elementary symmetric polynomials of the eigenvalues of A. Using Newton identities, the elementary symmetric polynomials can in turn be expressed in...

formula expressed in terms of partial (or incomplete) exponential Bell polynomials B n , k ( x 1 , … , x n − k + 1 ) {\displaystyle B_{n,k}(x_{1},\ldots...

moments, given by the polynomials above.[clarification needed][citation needed] For those polynomials, construct a polynomial sequence in the following...

of a partition Solid partition Young tableau Young's lattice Bell number Bell polynomials Dobinski's formula Cumulant Data clustering Equivalence relation...

represented in terms of traces of powers of A using complete exponential Bell polynomials. The resulting formula is adj ⁡ ( A ) = ∑ s = 0 n − 1 A s ∑ k 1 , k...

s_{l}=-{\tfrac {1}{2}}(l-1)!\,\mathrm {tr} ((AB)^{l})} and Bn(s1,s2,...,sn) are Bell polynomials. For a block-diagonal matrix A 1 ⊕ A 2 = [ A 1 0 0 A 2 ] , {\displaystyle...

_{l=1}^{n-1}lk_{l}=n-1.} The formula can be rewritten in terms of complete Bell polynomials of arguments t l = − ( l − 1 ) ! tr ⁡ ( A l ) {\displaystyle t_{l}=-(l-1)...

other than numbers; for instance the Bell polynomials form a triangular array in which each array entry is a polynomial. Arrays in which the length of each...

generating function is given implicitly through the Bell polynomials by the EGF for these polynomials defined in the previous formula for some sequence...

consequence of the general relation between Z n {\displaystyle Z_{n}} and Bell polynomials: Z n ( x 1 , … , x n ) = 1 n ! B n ( 0 ! x 1 , 1 ! x 2 , … , ( n −...

then an explicit form of inverse coefficients can be given in term of Bell polynomials: g n = 1 f 1 n ∑ k = 1 n − 1 ( − 1 ) k n k ¯ B n − 1 , k ( f ^ 1 ,...

Triangular pyramidal number The (incomplete) Bell polynomials from a triangular array of polynomials (see also Polynomial sequence). Heronian triangle Integer...

went on to give the general solution for this problem in terms of the Bell polynomials, showing the traditional score overpredicts P-values by orders of magnitude...

distribution of match counts of pairs of integer multisets in terms of Bell polynomials, a problem directly relevant to physical mapping of DNA. Prior to this...

_{k=1}^{n}a_{k}^{j_{k}}} that can be also stated in terms of complete Bell polynomials: Z ( S n ) = B n ( 0 ! a 1 , 1 ! a 2 , … , ( n − 1 ) ! a n ) n ! ....

{1}{j^{k}j!}}} . Bell polynomials Catalan number Cycles and fixed points Pochhammer symbol Polynomial sequence Touchard polynomials Stirling permutation...

of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function...

x^{k}} from the following formal power series (see the non-exponential Bell polynomials and section 3 of ). More generally, sums related to these weighted...

Stirling numbers of the first kind Bell number – the number of partitions of a set with n members Stirling polynomials Twelvefold way Learning materials...