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Longer titles found: Three-term recurrence relation (view)

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Stirling numbers of the second kind (4,313 words) [view diff] exact match in snippet view article find links to article

entries would all be 0. Stirling numbers of the second kind obey the recurrence relation (first discovered by Masanobu Saka in his 1782 Sanpō-Gakkai): { n

given expression with the determinant. The polynomials Pn satisfy a recurrence relation of the form P n ( x ) = ( A n x + B n ) P n − 1 ( x ) + C n P n −

is the case for Gaussian quadrature), the recurrence relation reduces to a three-term recurrence relation: For s < r − 1 , x p s {\displaystyle s<r-1

<\pi .} The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relation ( n + 1 ) P n + 1 ( λ ) ( x ; ϕ ) = 2 ( x sin ⁡ ϕ + ( n + λ ) cos

Cardoso (2016). The Hahn–Exton q-Bessel function has the following recurrence relation (see Swarttouw (1992)): J ν + 1 ( 3 ) ( x ; q ) = ( 1 − q ν x + x

The Pomeau–Manneville map is a polynomial mapping (equivalently, recurrence relation), often referred to as an archetypal example of how complex, chaotic

{\displaystyle h_{n}(x;q)} satisfy (for n ≥ 1 {\displaystyle n\geq 1} ) the recurrence relation h n + 1 ( x ; q ) = ( 1 + x ) h n ( x ; q ) + x ( q n − 1 ) h n −

(2n+1)}}\delta _{mn}.} The sequence of Bateman polynomials satisfies the recurrence relation ( n + 1 ) 2 F n + 1 ( z ) = − ( 2 n + 1 ) z F n ( z ) + n 2 F n −

the root condition and means that the parasitic solutions of the recurrence relation will not grow exponentially. The following third-order method has

) {\displaystyle \phi _{n}(z)} for the polynomials, they obey a recurrence relation ϕ n + 1 ( z ) = z ϕ n ( z ) + ρ n + 1 ϕ n ∗ ( z ) {\displaystyle

In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex

Subtracting the first two expressions for the derivative gives the recurrence relation, z U ( a , z ) = U ( a − 1 , z ) − ( a + 1 2 ) U ( a + 1 , z ) .

harmonic number. The Bickley functions also satisfy the following recurrence relation: n Ki n + 1 ⁡ ( x ) = ( n − 1 ) Ki n − 1 ⁡ ( x ) − x Ki n ⁡ ( x )

1 ) {\displaystyle (f_{-1},f_{0},\dotsc ,f_{d-1})} by using the recurrence relation h 0 i = 1 , − 1 ≤ i ≤ d {\displaystyle h_{0}^{i}=1,\qquad -1\leq

{\text{Equation (2)}}} for all n ≥ 2. {\displaystyle n\geq 2.} This is a recurrence relation giving W n {\displaystyle W_{n}} in terms of W n − 2 {\displaystyle

the model is in the form of a stochastic difference equation (or recurrence relation) which should not be confused with a differential equation. Together

into several identical parts by removing a vertex. This gives us a recurrence relation which allows us to compute the magnetization of a Cayley tree with

positive as long as K < 1 {\displaystyle K<1} . P2. When expanding the recurrence relation, one obtains a formula for θ n {\displaystyle \theta _{n}} : θ n

133122, 122331. The Eulerian numbers of the second order satisfy the recurrence relation, that follows directly from the above definition: ⟨ ⟨ n k ⟩ ⟩ = (

second groups, G(N) = g(N, M ) = XM g(N -1,M ) + g(N,M -1) This same recurrence relation clearly exists for any intermediate value of n from 1 to N, and for

below) - the coefficients of the generalized power series obey a recurrence relation, so that they can always be straightforwardly calculated. A second

show that generalized Marcum Q-function satisfies the following recurrence relation Q ν + 1 ( a , b ) − Q ν ( a , b ) = ( b a ) ν e − ( a 2 + b 2 ) /

general (under weaker conditions), the convergence rate is linear. The recurrence relation for Newton's method for minimizing a function S of parameters β {\displaystyle

the points (xk, yk) for k = i, i + 1, ..., j. The pi,j satisfy the recurrence relation This recurrence can calculate p0,n(x), which is the value being sought

o m p ( n , r ) {\displaystyle Comp(n,r)} . Then, the following recurrence relation holds: C o m p ( n , r ) = C o m p ( n , r − 1 ) + C o m p ( n −

section "Generalization". Firstly, we confirm the validity of the recurrence relation C m ( n , k ) = C m ( n − 1 , k ) + C m ( n , k − 1 ) {\displaystyle

in Narayana’s cows sequence: 1,1,1,2,3,4,6,9,13,19,... with the recurrence relation Nn = Nn−1 +Nn−3 that starts at N0. It has been observed that the

common multiple of their denominators. Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations

in Narayana’s cows sequence: 1,1,1,2,3,4,6,9,13,19,... with the recurrence relation Nn = Nn−1 +Nn−3 that starts at N0. It has been observed that the

that the gamma function is the unique solution to the factorial recurrence relation that is positive and logarithmically convex for positive z and whose

\end{aligned}}} The Kravchuk polynomials satisfy the three-term recurrence relation x K k ( x ; n , q ) = − q ( n − k ) K k + 1 ( x ; n , q ) + ( q (

This latter form is the mathematical definition of factorial as a recurrence relation. Note that the compiler inferred the type of this function to be

{\displaystyle N^{-1+\varepsilon }} above. The recurrence relation above is similar to the recurrence relation used by a linear congruential generator, a

that the number of k-colorings is a polynomial in k follows from a recurrence relation called the deletion–contraction recurrence or Fundamental Reduction

{1}{z^{2n}}}\right).} Since these two sequences of polynomials satisfy the same recurrence relation in three terms, we can develop a continued fraction for the Stieltjes

be approximated by a discrete system, which leads to a following recurrence relation analog to the Hamilton–Jacobi–Bellman equation: J k ∗ ( x n − k )

Donald Knuth defined the more general Stirling numbers by extending a recurrence relation to all integers. In this approach, [ n k ] {\textstyle \left[{n \atop

5(1939)401-417. In it he finds a difference equation for a coefficient in the recurrence relation for polynomials orthogonal on the real line with respect to e^(-x^4)

equation, too (this is only true in machine precision). Thus is a recurrence relation for the error. Equations (1) and (2) show that both the error and

postorder traversal of a balanced binary tree. In addition, the recurrence relation would be simpler. But I know why I chose the Leonardo numbers: (transcription)

The values of a Krawchouk matrix can also be calculated using a recurrence relation. Filling the top row with ones and the rightmost column with alternating

expected number of coin tosses before a result is returned. The recurrence relation of f b ( p ) {\displaystyle f_{b}(p)} can be described as follows

number of numbers the formula can produce is the modulus, m. The recurrence relation can be extended to matrices to have much longer periods and better

{\displaystyle s} . Then the values of P {\displaystyle P} are given by the recurrence relation P t , s = { π s ⋅ b s , o t if  t = 0 , max r ∈ S ( P t − 1 , r ⋅

the number of (male) children of the jth of these descendants. The recurrence relation states that the number of descendants in the n+1st generation is

worth n ! 2 n + 1 . {\displaystyle {\frac {n!}{\sqrt {2n+1}}}.} The recurrence relation in three terms is written: 2 ( 2 n + 1 ) X P n ( X ) = − P n + 1

{\displaystyle {\mathcal {N}}_{n}(z)} satisfies the following nonlinear recurrence relation: N n ( z ) = ( 1 + z ) N n − 1 ( z ) + z ∑ k = 1 n − 2 N k ( z )

{x^{7}}{7^{n}}}+\cdots } is defined analogously. This satisfies the recurrence relation: Ti n ⁡ ( x ) = ∫ 0 x Ti n − 1 ⁡ ( t ) t d t {\displaystyle \operatorname

= k Δ t / h 2 . {\displaystyle r=k\Delta t/h^{2}.} So, with this recurrence relation, and knowing the values at time n, one can obtain the corresponding

1 − x {\displaystyle L_{1}(x)=1-x} and then using the following recurrence relation for any k ≥ 1: L k + 1 ( x ) = ( 2 k + 1 − x ) L k ( x ) − k L k

to Pascal's triangle. The first eight rows are shown below. The recurrence relation ( n k ) F = F n − k + 1 ( n − 1 k − 1 ) F + F k − 1 ( n − 1 k ) F

uv} is merely removed. Then the Tutte polynomial is defined by the recurrence relation T G = T G − e + T G / e , {\displaystyle T_{G}=T_{G-e}+T_{G/e},}

and similar cellular automata Cellular Automata FAQ – Conway's Game of Life Algebraic formula, recurrence relation for iterating Conway's Game of Life.

1,\dots ,n-1} . The monic orthogonal Szegő polynomials satisfy a recurrence relation of the form Φ n + 1 ( z ) = z Φ n ( z ) − α ¯ n Φ n ∗ ( z ) {\displaystyle

sequence can be extended to negative index n using the rearranged recurrence relation F n − 2 = F n − F n − 1 , {\displaystyle F_{n-2}=F_{n}-F_{n-1},}

effort of computing the result for one grid point. The following recurrence relation is then obtained for the effort of obtaining the solution on grid

distribution (which gives a linear function) and, secondly, from a recurrence relation for values in the probability mass function of the hypergeometric

) {\displaystyle q_{n}=P(R_{n})} , we get the linear homogeneous recurrence relation q n = q n + 1 p + q n − 1 q , {\displaystyle q_{n}=q_{n+1}p+q_{n-1}q

_{\varphi \varphi }} . The Zernike polynomials satisfy the following recurrence relation which depends neither on the degree nor on the azimuthal order of

-\alpha -(\alpha +\beta +2)x)y'+n(n+\alpha +\beta +1)y=0.} The recurrence relation for the Jacobi polynomials of fixed α {\displaystyle \alpha } , β

( z ) f ( z ) {\displaystyle zf^{\prime }(z)=g(z)f(z)} gives the recurrence relation ( n − 1 ) a n = ∑ k = 1 n − 1 b n − k a k . {\displaystyle (n-1)a_{n}=\sum

Improvements to overcome unwanted zero terms include an altered recurrence relation suggested by Jaaskelainen and Ruuskanen in 1981 or a simple shift

{b}}} . The sequence of continuous Hahn polynomials satisfies the recurrence relation x p n ( x ) = p n + 1 ( x ) + i ( A n + C n ) p n ( x ) − A n − 1

applications of cosine can be replaced with a single application and use of recurrence relation for Chebyshev polynomials. D = R θ 1 2 + θ 2 2 − 2 θ 1 θ 2 cos ⁡

A_{i}={\begin{pmatrix}s_{i-1}&s_{i}\\t_{i-1}&t_{i}\end{pmatrix}}.} The recurrence relation may be rewritten in matrix form A i + 1 = A i ⋅ ( 0 1 1 − q i )

expansions of (x)n,f,t and then by the next corresponding triangular recurrence relation: [ n k ] f , t = [ x k − 1 ] ( x ) n , f , t = f ( n − 1 ) t 1 −

base of the recursion and find C {\displaystyle C} is given by the recurrence relation P ( n ) = 1 − ( 1 − 1 2 P ( ⌈ 1 + n 2 ⌉ ) ) 2 {\displaystyle P(n)=1-\left(1-{\frac

{\displaystyle \varphi ([X,Y])=X\varphi (Y)-Y\varphi (X)} gives a recurrence relation for the a n {\displaystyle a_{n}} : ( m − n ) a m + n = ( m + λ n

{\frac {\color {orangered}\tau }{n}}} n-ball and n-sphere volume recurrence relation V n ( r ) = r n S n − 1 ( r ) {\displaystyle V_{n}(r)={\frac {r}{n}}S_{n-1}(r)}

based on an auxiliary function B(r,s), defined by the following recurrence relation: B ( r , s ) := 1     if     s ≤ 0 ; {\displaystyle B(r,s):=1~~{\text{if}}~~s\leq

As discussed above, the Legendre polynomials obey the three-term recurrence relation known as Bonnet's recursion formula given by ( n + 1 ) P n + 1 (

"Asymptotic Behaviour of Christoffel–Darboux Kernel Via Three-Term Recurrence Relation I". Constructive Approximation. 54 (1): 49–116. arXiv:1909.09107

2 {\displaystyle q'={\sqrt {r^{2}+q^{2}}}} . Thus, we obtain the recurrence relation G D + 2 ( t , r ) = − 1 2 π r ∂ r G D ( t , r ) . {\displaystyle

Z-transforms. Advanced Z-transform Bilinear transform Difference equation (recurrence relation) Discrete convolution Discrete-time Fourier transform Finite impulse

a:(1 − a). Therefore the error function satisfies the following recurrence relation: E ( m ) = 0          if  m ≤ b {\displaystyle E(m)=0\ \ \ \ {\text{

k 2 , … {\displaystyle k_{0},k_{1},k_{2},\dots } satisfies the recurrence relation k − 1 = 0 , k 0 = 1 , k n = k n − 1 a n + k n − 2 {\displaystyle

\omega )=\operatorname {fnchypg} (m_{1}-x;N-n,m_{1},N,1/\omega )\,.} Recurrence relation: fnchypg ⁡ ( x ; n , m 1 , N , ω ) = fnchypg ⁡ ( x − 1 ; n , m 1

method for computing Gaussian quadrature rules given the three term recurrence relation that the underlying orthogonal polynomials satisfy. They reduce the

}}{(2\ell +1)!}}+(-1)^{n+1}{\frac {n}{(2n+1)!}}\end{aligned}}} This recurrence relation may be derived from that for the Bernoulli numbers. Also, there is

{1}{x(x-1)\cdots (x-n)}}.} It follows that these polynomials satisfy the next recurrence relation given by ( x + 1 ) σ n ( x + 1 ) = ( x − n ) σ n ( x ) + x σ n −

identity. The Bernoulli polynomials of the second kind satisfy the recurrence relation ψ n ( x + 1 ) − ψ n ( x ) = ψ n − 1 ( x ) {\displaystyle \psi _{n}(x+1)-\psi

the following table. The Akiyama–Tanigawa numbers satisfy a simple recurrence relation which can be exploited to iteratively compute the Bernoulli numbers

be corrected without knowing their locations. There is a linear recurrence relation that gives rise to a system of linear equations. Solving those equations

theory, studied by Euler, Hardy, Ramanujan, Erdős, and others. The recurrence relation P ( n ) = ∑ k = 1 n ( − 1 ) k + 1 [ P ( n − 1 2 k ( 3 k − 1 ) ) +

the procedure proceeds recursively. This leads to the following recurrence relation (where k is the number of agents in P, not including the clone of

sequences of orthogonal polynomials can always be given a three-term recurrence relation.) For k = j − 1 {\displaystyle k=j-1} one gets h j − 1 , j = ( A

of the reciprocal generating function, then we have the following recurrence relation: b n = − 1 f 0 ( f 1 b n − 1 + f 2 b n − 2 + ⋯ + f n b 0 ) , n ≥

f:\mathbb {N} \rightarrow \mathbb {C} } that satisfies a linear recurrence relation: that is, if a 1 f ( n + b 1 ) + ⋯ + a k f ( n + b k ) = 0 {\displaystyle

{1}{2}}\end{array}}} Generalized Stieltjes constants satisfy the following recurrence relation γ n ( a + 1 ) = γ n ( a ) − ( ln ⁡ a ) n a , n = 0 , 1 , 2 , … a

{\displaystyle \varphi } is transcendental, so the definition is a recurrence relation. The initial guess φ {\displaystyle \varphi } is a small fraction

{\displaystyle m} . The first step is to obtain an algebraic expansion of the recurrence relation corresponding to each sailor's transformation of the pile, n i {\displaystyle

+\lambda e^{-\phi })} Although its length varies directly with n, this recurrence relation is only employed for numerical computation and is particularly useful