mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Racah polynomials are

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product. The most widely used orthogonal polynomial ...

named after

Giulio Racah Giulio (Yoel) Racah ( he, ג'וליו (יואל) רקח; February 9, 1909 – August 28, 1965) was an Italian–Israeli physicist and mathematician. He was Acting President of the Hebrew University of Jerusalem from 1961 to 1962. The crater ...

, as their orthogonality relations are equivalent to his orthogonality relations for

Racah coefficient Wigner's 6-''j'' symbols were introduced by Eugene Paul Wigner in 1940 and published in 1965. They are defined as a sum over products of four Wigner 3-j symbols, : \begin j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end = \sum_ (-1)^ \beg ...

s. The Racah polynomials were first defined by and are given by :

p_n(x(x+\gamma+\delta+1)) = _4F_3\left begin -n &n+\alpha+\beta+1&-x&x+\gamma+\delta+1\\
\alpha+1&\gamma+1&\beta+\delta+1\\ \end;1\right

Orthogonality

\sum_^N\operatorname_n(x;\alpha,\beta,\gamma,\delta) 
\operatorname_m(x;\alpha,\beta,\gamma,\delta)\frac \omega_y=h_n\operatorname_,

:when

\alpha+1=-N

, :where

\operatorname

is the Racah polynomial, :

x=y(y+\gamma+\delta+1),

\operatorname_

is the

Kronecker delta function In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\t ...

and the weight functions are :

\omega_y=\frac,

:and :

h_n=\frac\frac\frac,

(\cdot)_n

is the

Pochhammer symbol In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \e ...

Rodrigues-type formula

\omega(x;\alpha,\beta,\gamma,\delta)\operatorname_n(\lambda(x);\alpha,\beta,\gamma,\delta)=(\gamma+\delta+1)_n\frac\omega(x;\alpha+n,\beta+n,\gamma+n,\delta),

:where

\nabla

is the

backward difference operator A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...

, :

\lambda(x)=x(x+\gamma+\delta+1).

Generating functions

There are three generating functions for

x\in\

:when

\beta+\delta+1=-N\quad

\quad\gamma+1=-N,

_2F_1(-x,-x+\alpha-\gamma-\delta;\alpha+1;t)_2F_1(x+\beta+\delta+1,x+\gamma+1;\beta+1;t)

\quad=\sum_^N\frac\operatorname_n(\lambda(x);\alpha,\beta,\gamma,\delta)t^n,

:when

\alpha+1=-N\quad

\quad\gamma+1=-N,

_2F_1(-x,-x+\beta-\gamma;\beta+\delta+1;t)_2F_1(x+\alpha+1,x+\gamma+1;\alpha-\delta+1;t)

\quad=\sum_^N\frac\operatorname_n(\lambda(x);\alpha,\beta,\gamma,\delta)t^n,

:when

\alpha+1=-N\quad

\quad\beta+\delta+1=-N,

_2F_1(-x,-x-\delta;\gamma+1;t)_2F_1(x+\alpha+1;x+\beta+\gamma+1;\alpha+\beta-\gamma+1;t)

\quad=\sum_^N\frac\operatorname_n(\lambda(x);\alpha,\beta,\gamma,\delta)t^n.

Connection formula for Wilson polynomials

When

\alpha=a+b-1,\beta=c+d-1,\gamma=a+d-1,\delta=a-d,x\rightarrow-a+ix,

\operatorname_n(\lambda(-a+ix);a+b-1,c+d-1,a+d-1,a-d)=\frac,

:where

\operatorname

are Wilson polynomials.

q-analog

introduced the ''q''-Racah polynomials defined in terms of

basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called h ...

s by :

p_n(q^+q^cd;a,b,c,d;q) = _4\phi_3\left begin q^ &abq^&q^&q^cd\\
aq&bdq&cq\\ \end;q;q\right

They are sometimes given with changes of variables as :

W_n(x;a,b,c,N;q) = _4\phi_3\left begin q^ &abq^&q^&cq^\\
aq&bcq&q^\\ \end;q;q\right

References

* * Orthogonal polynomials {{algebra-stub