This chapter focuses on the hypergeometric orthogonal polynomials within the Askey scheme, detailing their most significant properties for each family. These properties include representations as hypergeometric functions, orthogonality relations, three-term recurrence relations, second-order differential or difference equations, forward and backward shift operators, Rodrigues-type formulas, and generating functions. The chapter uses common notations from the literature and highlights connections between different families through limit relations. For the Wilson polynomials, a specific hypergeometric representation is provided:
\[
\frac{W_{h}(x^{2} ; a, b, c, d)}{(a+b)_{n}(a+c)_{n}(a+d)_{n}} =_{4} F_{3}(\begin{array}{c}
-n, n+a+b+c+d-1, a+i x, a-i x \\
a+b, a+c, a+d
\end{array} ; 1).
\]This chapter focuses on the hypergeometric orthogonal polynomials within the Askey scheme, detailing their most significant properties for each family. These properties include representations as hypergeometric functions, orthogonality relations, three-term recurrence relations, second-order differential or difference equations, forward and backward shift operators, Rodrigues-type formulas, and generating functions. The chapter uses common notations from the literature and highlights connections between different families through limit relations. For the Wilson polynomials, a specific hypergeometric representation is provided:
\[
\frac{W_{h}(x^{2} ; a, b, c, d)}{(a+b)_{n}(a+c)_{n}(a+d)_{n}} =_{4} F_{3}(\begin{array}{c}
-n, n+a+b+c+d-1, a+i x, a-i x \\
a+b, a+c, a+d
\end{array} ; 1).
\]