Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions by P.L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials.

In L. N. Trefethen' s book, Approximation theory and approximation practice, page 271.

It was Waring in 1779, Euler used the formula in 1783, and Lagrange in 1795.

The explicit formula of the Lagrange interpolating polynomial was first published by Waring (1779), rediscovered by Euler in 1783, and published by Lagrange in 1795 (cf. Jeffreys, H. and Jeffreys, B. S. "Lagrange's Interpolation Formula." §9.011 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 260, 1988.)

With respect to the general theory of orthogonal polynomials really started with the investigations of Chebyshev and Stieltjes (see for instance, The impact of Stieltjes' work on continued fractions and orthogonal polynomials by Walter Van Assche, and the references therein. http://arxiv.org/pdf/math/9307220v1.pdf)

Initially, mathematician introduced orthogonal polynomial without having a theory of orthogonal polynomial. Do you want to know who introduced the first polynomial known now to an orthogonal family (the classical polynomials are orthogonal in some sens) or who produced the first description with a notion of orthogonality on polynomials (in this case it is perhaps Carl Jacobi) ?

Dear @Olivier, Thank you for the precise answer.

Dears @Qefsere, @Hua, and @Yamilet, thank you very much for the detailed answers.

Followers of this thread may find the Wolfram Mathworld article on orthogonal polynomials interesting, especially with its list of references to foundation papers:

http://mathworld.wolfram.com/OrthogonalPolynomials.html

Dear @James, Thank you very much for the information and the article

Weisstein, Eric W. "Orthogonal Polynomials." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OrthogonalPolynomials.html

It contains some of the properties of the classical orthogonal polynomials.

Dear Professor Abedallah M Rababah,

The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was pursued by A.A. Markov and T.J. Stieltjes. Some of the mathematicians who have worked on orthogonal polynomials include Gábor Szegő, Sergei Bernstein, Naum Akhiezer, Arthur Erdélyi, Yakov Geronimus, Wolfgang Hahn, Theodore Seio Chihara, Mourad Ismail, Waleed Al-Salam, and Richard Askey [1].

[1] http://en.wikipedia.org/wiki/Orthogonal_polynomials

Dear @Professor Afag Ahmad, your contribution is highly appreciated. I want to know in particular who did first define orthogonal polynomials.