mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ...

named after

Giulio Racah Giulio (Yoel) Racah (; 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 Racah on the Moon is named after hi ...

, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients. 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 &\te ...

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) . \end ...

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, :

\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 hy ...

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