A CLASS OF ORTHOGONAL POLYNOMIALS OF A NEW TYPE

The purpose of this paper is to study a class of polynomials Fnm(x,λ,ν) associated with a class of m differential equations, where λ is fixed, n is a variable parameter, m is fixed positive integer and ν is a non-negative integer <m. We also obtain all the important properties of this class of polynomials including Rodrigues' formulas, generating function, and recurrence relations. Several special cases of
interest are obtained from our analysis.

Similar generalization of the class of Hermite polynomials, so that two differ- ential equations, two Rodrigues' formulas and two recurrence relations for even and odd values of the variable parameter and associated in the same function space, was obtained, and studied in some details by Dutta, Chatterjea, Ghosh [3] and Dutta, Chatterjea and More [4].Earlier in the paper of Krall and Frink  [5], some similar result i.e. two different differential equations in the discussion of a class of funct- ions written by suitable modification of Bessel functions are seen, but this pecular- ity is not studied with due emphasis.
Theoretical justification of the association of a pair of differential equations with a single class of orthogonal functions has been discussed in some details in Dutta and Bhattacharya [6].In a subsequent paper of Dutta [7], the possibility of a class of orthogonal polynomials having m number of differential equations, m recurrence relations, etc. (m being a.fixed natural number) is established.
In this paper following the suggestions of Dutta [7], it is shown how and when solutions of m differential equations may belong to the same class of orthogonal poly- nomials.This new class of orthogonal polynomials satisfies m Rodrigues' formulas, m recurrence relations, etc., where m being the number of partitions of the variable parameter n.
Let us consider a class of m differential equations: x 2 x where is fixed, n is a variable parameter, m is a fixed positive integer where is a non-negative integer < m.
Here we see that for different values of v we shall obtain m different different- ial equations, but solutions of these differential equations are orthogonal in the w(x,%)(i) where same function space L 2 m -x +m-2 w(x,%) e Ixl I (0, ) when m is odd, (_oo,m) when m is even.
Of course for an odd m, as I is (0, ) so Ix is obviously replaced by x.The in- tervals are taken from the consideration of convergence of the integral defining or tho gonali ty.
2. THE CLASS OF POLYNOMIALS {Fm(x,l )} AND THEIR ORTHOGONALITY.n Solving the differential equation (I.i), we obtain a class of functions n-u for running parameter n and for fixed %, m the functions will reduce to polynomials for n (mod m) and = 0,1,...,m-l, where is a non-negative integer < m.Now all the polynomials for different values of n with n (mod m) form a class of orthogo- nal polynomials.
For differnet values of corresponding to a particular (fixed) value of m we shall obtain m different classes of polynomials.
ORTHOGONALITY.Following the technique of Dutta and Bhattacharya [6] and Dutta m -x l+m-2 [7], we can calculate the weight function w(x,l) e Ixl of the function space properly associated with the class of differential equations and establishes the or- thogonality relation: m e -x Ixlx+m-2 F m (x,X ) F,(x,X, ) dx 6 i j ij (2.2) where I (0,o) if m is odd, (-o%00) if m is even.

THE POLYNOMIALS AS 2F0
From the equation (2.1) by reversing the order of summation, we can write at once In (2.1) replacing x by --(here we are taking the real positive root of /) and n by mn+ , we obtain where n (rood m).
This can be written in terms of Laguerre polynomials: 5. RODRIGUES FORMULAS.
For Laguerre polynomials, Rodrigues' formula is given by L (a) ( Obviously, for different values of v ( 0,1,2, m-l), we shall obtain m diff- erent Rodrigues' formulas.
6. GENERATING FUNCTIONS.Equation (2.1) can be written as: Multiplying both sides by .and taking the sum from n 0 to oo, tn n (-n) Thus the generating function for Fm(x,% ) is given by the relation To find out the other generating function multiplying both sides of the equation (6.1) t n by (C)n and taking the sum from 0 to oo, With the choice c 2 +l+m-i 2+l+m-I equation (6.3) can be written as: )n 'mn+V (x'l') (i t) x 0F0( m (i t) Hence our required generating function is given by 2+l+m- )n mn+ .
In this section, we obtain pure and mixed recurrence relations for the polynomials Fm(x, ).n 7.1.DIFFERENTIAL RECURRENCE RELATIONS.In the last section, we have obtained the generating function for the polynomials Fm(x,% ) given by ( 6 e 0FI(--; x t) x e (--; -x t), (7.4) t m m where prime denotes the differentiation with respect to the argument.from (7 2), (7 3) and (7.4), we see that G(x,t) satisfies Eliminating 0FI, 0FI the following partial differential equation:

G 8G
x x m t m t G. (7.5) Substituting G from (7.1)To find out the other differential recurrence relation, let us start from the generating relation given by (6.4), which can be written as: Writing G in place of G(x,t), and differentiating (7.9) partially with respect to x: Substituting G from (7.8), we obtain Equating the coefficients of t n from both sides, we obtain the differential recurrence relation  ) Fm m(n_l)+(x,%,) + n + m(n i m(n_l)+(x,%,) Hence we obtain ran(ni) F m m(n_2)+v(x,%, (n +2+ %-1 F TM 1 m m mn+V (x' ) n 1 + 2v+ % m x F m m(n_l)+o(x % ) + (n-l) F m m(n_2)+(x,,) 0. (7.14)In the above equation we see that for a fixed m and for different values of 0,1,2,...,m-1), we shall obtain m different recurrence relations.Also we are getting different recurrence relations for different partitions for the set of values of the variable parameter n.For example if we take m 2, then for v 0 (7.14) gives us the recurrence relation for even polynomials and for 1 it gives us that for odd polynomials, which is interesting and seems to be new.where L () (x) is the Laguerre polynomials.
But for % i, i.e. simple Laguerre polynomials.
With the help of these relations, we can calculate generating functions, recurrence relations, etc. for the polynomials H2n(X) and H2n+l(X).