https://doi.org/10.4134/CKMS.c200208 pISSN: 1225-1763 / eISSN: 2234-3024
A GENERALIZATION OF THE LAGUERRE POLYNOMIALS
Asad Ali
Abstract. The main aim of this paper is to introduce and study the generalized Laguerre polynomials and prove that these polynomials are characterized by the generalized hypergeometric function. Also we in-vestigate some properties and formulas for these polynomials such as explicit representations, generating functions, recurrence relations, dif-ferential equation, Rodrigues formula, and orthogonality.
1. Introduction and preliminaries
Laguerre polynomials are among the most important and useful polynomials in mathematics and mathematical physics. Most of monographs and books related to special functions include Laguerre polynomials (see, e.g., [3, 15, 16]). Laguerre polynomials L(α)n (x) are defined by (see, e.g., [15, Chapter 12])
(1) L(α)n (x) = (1 + α)n
n! 1F1(−n ; 1 + α ; x) n ∈ N0, 1 + α ∈ C \ Z−0, x ∈ C ,
where 1F1 is a particular case of the well-known generalized hypergeometric series pFq (p, q ∈ N0) given by (see, e.g., [15, p. 73]):
(2) pFq λ 1, . . . , λp; µ1, . . . , µq; z = ∞ X n=0 (λ1)n· · · (λp)n (µ1)n· · · (µq)n zn n! =pFq(λ1, . . . , λp; µ1, . . . , µq; z). Here (α)β denotes the Pochhammer symbol defined (for α, β ∈ C) by (3) (α)β:=
Γ(α + β)
Γ(α) =
(1 (β = 0; α 6= 0)
α(α + 1) · · · (α + n − 1) (β = n ∈ N),
Received June 18, 2020; Revised November 19, 2020; Accepted November 23, 2020. 2010 Mathematics Subject Classification. Primary 33C10, 33C20; Secondary 33C45, 44A10.
Key words and phrases. Laguerre polynomials, generalized polynomials, generalized hy-pergeometric series, generating functions, recurrence relations, differential equation, Ro-drigues formula, orthogonality.
c
2021 Korean Mathematical Society
Γ being the familiar Gamma function and it being read traditionally that (α)0:= 1. Here and elsewhere, let N, Z−0, R, and C denote the sets of positive integers, non-positive integers, real numbers, and complex numbers, respec-tively, and set N0:= N ∪ {0}. The particular case α = 0 of (1)
(4) Ln(x) = L(0)n (x) =1F1(−n ; 1 ; x) (n ∈ N0, x ∈ C)
is called as simple Laguerre (or Laguerre) polynomial which has also attracted much attention. For certain formulas and properties including these polyno-mials, one may be referred (for example) to [1], [3, Section 6.2], [5, 6, 12–14], [15, pp. 201–202], [7–11, 17, 18].
Among numerous generating functions which can produce (1) or (4), we recall the following (see, e.g., [15, p. 202])
(5) 1 (1 − t)1+α exp −xt 1 − t = ∞ X n=0 L(α)n (x) tn.
Ali et al. [2] brought in a generalization of Bateman polynomial and pre-sented some interesting and presumably useful properties and formulas involv-ing it. In the same vein, in this paper, we introduce a generalization of Laguerre polynomials and investigate certain properties and formulas associated with it such as recurrence relation, differential formula, generating function, Rodrigues formula, and orthogonality.
2. Generalized Laguerre polynomials
We begin by introducing generalized Laguerre polynomials, which are de-noted by L(α)p,n(x) whose generating function is given as in Definition 1. Definition 1. Let p ∈ N; x, α ∈ C. (6) 1 (1 − t)1+αexp −xptp (1 − t)p = ∞ X n=0 L(α)n,p(x) tn (p ∈ N; x, α ∈ C) .
Obviously L(α)n,1(x) = L(α)n (x). Hereafter we explore certain formulas and properties involving the generalized Laguerre polynomials in (6). Throughout, let F (p; x, t) be the left-handed generating function in (6).
Explicit representation
An explicit expression of the generalized Laguerre polynomials L(α)n,p(x) in the following theorem.
Theorem 2.1. Let x, α ∈ C, p ∈ N, and n ∈ N0. Then L(α)n,p(x) = (1 + α)n [n/p] X k=0 (−1)k k! (1 + α)pk(n − pk)! xpk (7) =(1 + α)n n! [n/p] X k=0 (−1)(p+1)k(−n)pk k! (1 + α)pk xpk. (8)
Here and throughout, [m] denotes the greatest integer less than or equal to m ∈ R. Or, equivalently, (9) L(α)n,p(x) = (1 + α)n n! pFp −n p , −n + 1 p , . . . , −n − 1 + p p ; α + 1 p , α + 2 p , . . . , α + p p ; (−1)p+1xp .
Proof. Expanding the exponential in the left-hand side of (6), we find F (p; x, t) = 1 (1 − t)1+α+pk ∞ X k=0 (−1)kxpktpk k! .
Employing the binomial theorem (10) (1 − z)−a= ∞ X n=0 (a)n n! z n= 1F0(a ; ; z) (a ∈ C; |z| < 1),
we obtain the following double series (11) F (p; x, t) = ∞ X n=0 ∞ X k=0 (−1)k(1 + α + pk) nxpk k! n! t n+pk.
Recall a known double series manipulation (see, e.g., [4, Eq. (1.1)])
(12) ∞ X n=0 ∞ X k=0 Ak,n= ∞ X n=0 [n/p] X k=0 Ak,n−pk (p ∈ N) ⇐⇒ (13) ∞ X n=0 [n/p] X k=0 Ak,n= ∞ X n=0 ∞ X k=0 Ak,n+pk (p ∈ N),
where Ax,y denotes a function of two variables x and y and the involved double series is assumed to be absolutely convergent.
An application of (12) in (11) gives (14) F (p; x, t) = ∞ X n=0 [n/p] X k=0 (−1)k(1 + α + pk) n−pkxpk k! (n − pk)! t n.
Equating the coefficients of tn in the right members of (6) and (14) yields (15) L(α)n,p(x) = [n/p] X k=0 (−1)k(1 + α + pk) n−pk k! (n − pk)! x pk.
Using (3) and a known identity
(16) (n − k)! = (−1) kn! (−n)k (k, n ∈ N 0; 0 ≤ k ≤ n) , we derive (17) (1 + α + pk)n−pk= (1 + α)n (1 + α)pk and (n − pk)! = (−1) pkn! (−n)pk . Hence, use of (17) in (15) leads to the desired identity (8).
Finally, applying the multiplication formula (18) (λ)mn= mmn m Y j=1 λ + j − 1 m n (λ ∈ C; m ∈ N; n ∈ N0)
to (8) provides the equivalent expression (9).
Remark 2.2. Eq. (8) reveals that, for each n ∈ N0, L (α)
n,p(x) is a polynomial in the variable x of degree at most p[n/p]. In fact, the degree of L(α)n,p(x) is a step function in the following manner:
(19) deg L(α)n,p(x) = `p (`p ≤ n < (` + 1)p; ` ∈ N0) . Generating function
Establish two generating functions for the generalized Laguerre polynomials L(α)n,p(x) in Theorem 2.3.
Theorem 2.3. Let t, x, α, c ∈ C and p ∈ N. Then
(20) et0Fp ; α + 1 p , α + 2 p , . . . , α + p p ; − xt p p = ∞ X n=0 L(α)n,p(x) tn (1 + α)n and (21) 1 (1 − t)c pFp c p, c+1 p , . . . , c+p−1 p ; α+1 p , α+2 p , . . . , α+p p ; − xt 1 − t p! = ∞ X n=0 (c)nL (α) n,p(x) tn (1 + α)n (|t| < 1).
Proof. Using (7), (13), and (18), we have (22) ∞ X n=0 L(α)n,p(x) tn (1 + α)n = ∞ X n=0 tn n! ∞ X k=0 (−xptp)k k! (1 + α)pk = et ∞ X k=0 1 k! p Q j=1 α+j p k − xt p pk .
In view of (2), the rightmost term of (22) can be expressed as the left-hand side of (20).
Employing (7), (13), and (10), we find ∞ X n=0 (c)nL (α) n,p(x) tn (1 + α)n = ∞ X k=0 ∞ X n=0 (c + pk)ntn n! · (c)pk {−(xt)p} k k! (1 + α)pk = 1 (1 − t)c ∞ X k=0 (c)pk k! (1 + α)pk − xt 1 − t pk , which, upon using (18) and (2), leads to the left-hand member of (21).
It is noted that the case c = 1 + α of (21) yields the generating function (6). Recurrence relation
Present some recurrence relations involving the generalized Laguerre poly-nomials L(α)n,p(x) and their derivative in the following theorem.
Theorem 2.4. Let x, α ∈ C and p, n ∈ N. Also let D = dxd. Then (23) x DL(α)n,p(x) − n L(α)n,p(x) + (α + n) L (α) n−1,p(x) = 0; (24) DL(α)n,p(x) = 0 (0 ≤ n ≤ p − 1) −p xp−1L(α+p) n−p,p(x) (n ≥ p); (25) (α + n) L(α)n−1,p(x) − n L(α)n,p(x) = p x p L(α+p)n−p,p(x) (n ≥ p). Proof. From (22), we can set
(26) G(p; x, t) := ∞ X n=0 L(α)n,p(x) tn (1 + α)n = etΦ − xt p p , where the function
Φ − xt p p = ∞ X k=0 1 k! p Q j=1 α+j p k − xt p pk .
Differentiating G(p; x, t) with respect to x and t, respectively, gives Gx(p; x, t) = etΦ 0 − xt p p · −x p−1tp pp−1 and Gt(p; x, t) = etΦ − xt p p + etΦ0 − xt p p · −x ptp−1 pp−1 . Combining Gx(p; x, t) and Gt(p; x, t) yields
(27) x Gx(p; x, t) − t Gt(p; x, t) + t G(p; x, t) = 0. Applying the series in (26) to (27), we obtain
(28) ∞ X n=1 x DL(α)n,p(x) tn (1 + α)n − ∞ X n=1 n L(α)n,p(x) tn (1 + α)n + ∞ X n=1 L(α)n−1,p(x) tn (1 + α)n−1 = 0. We observe from (28) that each coefficient of tn should be zero, which gives (23).
Differentiating both sides of (6) provides ∞ X n=1 DL(α)n,p(x) tn= 1 (1 − t)1+α+p exp −xptp (1 − t)p · −p xp−1tp = −p xp−1 ∞ X n=0 L(α+p)n,p (x) tn+p = −p xp−1 ∞ X n=p L(α+p)n−p,p(x) tn,
which, upon equating the coefficients of tn(n ≥ p) in the leftmost and rightmost members, produces (24).
Setting (24) in (23) provides (25).
Differential equation
Provide a differential equation which is satisfied by the generalized Laguerre polynomials L(α)n,p(x) in Theorem 2.5 (for differential equation whose solution is pFq, see, e.g., [15, Section 47]).
Theorem 2.5. Let x, α ∈ C and p, n ∈ N. Also let θ = xdxd. Then
(29) 1 pθ p Y j=1 1 p(θ/p − 1 + α + j) + (−1)pxp p Y j=1 1 p(θ + j − n − 1) L(α)n,p(x) = η(α, p, n),
where α+jp ∈ C \ Z−0 and no two α+j
p differ by an integer (j = 1, . . . , p), and
η(α, p, n) = (−1)p+(p+1)[n/p] p p(1 + α) n· (−n + p[n/p])p n! [n/p]! × p Q j=1 j−n−1 p [n/p] p Q j=1 j+α p [n/p] xp([n/p]+1).
Proof. We derive from (8) that
(30) L(α)n,p(x) = (1 + α)n n! [n/p] X k=0 p Q j=1 j−n−1 p k k! p Q j=1 j+α p k (−1)(p+1)kxpk. Since θ px pk= k xpk, we have (31) 1 p(θ/p + j + α − 1)x pk= k + j + α − 1 p x pk.
Applying (30) to the following differential operator with the aid of (31), we get
(32) LDE := 1 pθ p Y j=1 1 p(θ/p − 1 + α + j) L(α)n,p(x) = (1 + α)n n! [n/p] X k=1 p Q j=1 j−n−1 p k· k p Q j=1 k+j+α−1 p k! p Q j=1 j+α p k (−1)(p+1)kxpk.
We then obtain that
LDE= (1 + α)n n! [n/p] X k=1 p Q j=1 j−n−1 p k (k − 1)! p Q j=1 j+α p k−1 (−1)(p+1)kxpk.
Putting k − 1 = k0 and cancelling the prime on k provides
LDE= (−1)p+1xp (1 + α)n n! [n/p]−1 X k=0 p Q j=1 j−n−1 p k· p Q j=1 j−n−1 p + k k! p Q j=1 j+α p k (−1)(p+1)kxpk.
We get LDE= (−1)p+1xp (1 + α)n n! [n/p] X k=0 p Q j=1 j−n−1 p k · p Q j=1 j−n−1 p + k k! p Q j=1 j+α p k (−1)(p+1)kxpk + η(α, p, n). Noting p Y j=1 θ p+ j − n − 1 p xpk= p Y j=1 k +j − n − 1 p xpk,
we find from (30) that
(33) LDE=(−1)p+1xp p Y j=1 θ + j − n − 1 p L(α)n,p(x) + η(α, p, n).
Finally, matching the first equality of (32) with (33) gives (29). The Rodrigues formula
Here and throughout, let Dk= dk
dxk (k ∈ N0). We give the Rodrigues formula for the generalized Laguerre polynomials L(α)n,p(x) in the following theorem. Theorem 2.6. Let x, α ∈ C and p, n ∈ N. Then
(34) L(α)n,p(x) = x−α exp−(−1)p1x n! D nhexp(−1)p1xxn+αi, where n is a multiple of p. Proof. Here (7) is written:
(35) L(α)n,p(x) = [n/p] X k=0 (−1)k(1 + α)n k! (1 + α)pk(n − pk)! xpk. Noting Dn−pkxn+α= (1 + α)nx α+pk (1 + α)pk and Dpkexp(−1)1px = (−1)k exp(−1)1px ,
we may get L(α)n,p(x) = x−α exp−(−1)p1x n! × [n/p] X k=0 n!nDpkexp(−1)p1x o Dn−pkxn+α k! (n − pk)! = x−α exp−(−1)p1x n! × [n/p] X k=0 n pk n Dpkexp(−1)1px o Dn−pk xn+α = x−α exp−(−1)p1x n! D nhexp(−1)p1xxn+αi. Orthogonality
Explore orthogonality for the generalized Laguerre polynomials L(α)n,p(x) in Theorem 2.7.
Theorem 2.7. Let x, α ∈ C with <(α) > 1 and p, m, n ∈ N be such that p is odd. Then (36) Z ∞ 0 xαexp(−1)1px L(α)n,p(x) L(α)m,p(x) dx = 0 (m 6= n) . Also (37) Z ∞ 0 xαexp(−1)1px n L(α)n,p(x) o2 dx = (−1) n+[n/p] [n/p]! Γ(1 + α + n), where n is a multiple of p.
Proof. Let Lm,n(p; α) be the left member of (36). Applying the Rodrigues formula (35) gives Lm,n(p; α) = 1 n! Z ∞ 0 Dnhexp(−1)1px xn+αiL(α)m,p(x) dx. Integrating by parts n times, we obtain
(38) Lm,n(p; α) = (−1)n n! Z ∞ 0 exp(−1)1px xn+α hDnL(α)m,p(x)idx. In the process of integrating by parts, the integrated section
Dn−khexp(−1)1px
xn+αiDk−1hL(α)m,p(x)i (1 ≤ k ≤ n)
vanishes both at x = 0 and as x → ∞ when p is odd and <(α) > −1. Since L(α)m,p(x) is of degree at most m, DnL
(α)
that Lm,n(p; α) = 0 for n > m. Since the integral Lm,n(p; α) is symmetric in n and m, Lm,n(p; α) = 0 for n < m. This proves (36).
From (7), we have L(α)n,p(x) = (−1) [n/p](1 + α) n [n/p]! (1 + α)p[n/p](n − p[n/p])! xp[n/p]+ $n(x),
where $n(x) is a polynomial in x of degree at most p[n/p] − 1. In particular, when n is a multiple of p, L(α)n,p(x) = (−1) [n/p] [n/p]! x n+ $ n−1(x),
where $n−1(x) is a polynomial in x of degree at most n − 1. Therefore we find
(39) DnL(α)n,p(x) = (−1)
[n/p] [n/p]! n!,
where n is a multiple of p. Setting (39) in the case m = n of (38) yields Ln,n(p; α) = (−1)n+[n/p] [n/p]! Z ∞ 0 e−xxn+αdx = (−1) n+[n/p] [n/p]! Γ(1 + α + n) (<(α) > −1),
which p is an odd positive integer and n ∈ N is a multiple of p. Some other properties
Provide some other identities involving the generalized Laguerre polynomials L(α)n,p(x) in Theorem 2.8.
Theorem 2.8. Let x, y, α, β ∈ C and p, n ∈ N. Then
(40) Z ∞ 0 xαe−xL(α)p,n(x) dx =Γ(1 + α + n) n! ×pF0 −n p , −n + 1 p , . . . , −n + p − 1 p ; ; (−1) p+1 pp (<(α) > −1; n is a multiple of p) ; (41) L(α)n,p(x) = n X k=0 (α − β)kL (β) n−k,p(x) k! ; (42) L(α+β+1)n,p (z) = n X k=0 L(α)k,p(x) L(β)n−k,p(y),
where xp+ yp ∈ C \ {0} and z := (xp+ yp)1p whose principal branch can be chosen; (43) L(α)n,p(xy) = n X k=0 (1 + α)n(1 − y)n−kykL (α) k,p(x) (n − k)! (1 + α)k .
Proof. Using (8) and Euler’s integral of the gamma function with the aid of (3), we have Z ∞ 0 xαe−xL(α)p,n(x)dx = (1 + α)n n! [n/p] X k=0 (−1)(p+1)k(−n)pk k! (1 + α)pk Z ∞ 0 e−xxα+pkdx = Γ(1 + α + n) n! [n/p] X k=0 (−1)(p+1)k(−n) pk k! ,
which, upon using (18) and (2), yields (40). From (6), we have ∞ X n=0 L(α)n,p(x) tn = (1 − t)−1−αexp −xptp (1 − t)p = (1 − t)−(α−β)· (1 − t)−1−βexp −xptp (1 − t)p = ∞ X n=0 ∞ X k=0 (α − β)k k! L (β) n,p(x)t n+k = ∞ X n=0 n X k=0 (α − β)k k! L (β) n−k,p(x)t n ,
which, upon equating the coefficients of tn, yields (41). We find from (6) that
∞ X n=0 n X k=0 L(α)k,p(x) L(β)n−k,p(y) tn = (1 − t)−1−αexp −xptp (1 − t)p (1 − t)−1−βexp −yptp (1 − t)p = (1 − t)−1−(α+β+1) exp −zptp (1 − t)p = ∞ X n=0 L(α+β+1)n,p (z) tn,
We consider et0Fp ; α + 1 p , α + 2 p , . . . , α + p p ; −xyt p p = e(1−y)teyt0Fp ; α + 1 p , α + 2 p , . . . , α + p p ; −x(yt) p p , which, in view of (20), produces
∞ X n=0 L(α)n,p(xy) tn (1 + α)n = ∞ X n=0 (1 − y)ntn n! ! ∞ X k=0 L(α)k,p(x) yktk (1 + α)k ! .
Then, from the last equality, we obtain (43).
3. Conclusion remarks
Since L(α)n,1(x) = L(α)n (x) and L(0)n,1(x) = Ln(x), the results in Section 2 reduce to yield certain properties and formulas for the Laguerre polynomials L(α)n (x) and the simple Laguerre polynomials Ln(x).
The identity (9) is rewritten as follows:
(44) n! (1 + α)n L(α)n,p(x) =pFp −n p , −n + 1 p , . . . , −n − 1 + p p ; α + 1 p , α + 2 p , . . . , α + p p ; (−1) p+1xp , which, upon setting p = 1, yields a known expression for the Laguerre polyno-mials
(45) n!
(1 + α)n
L(α)n (x) =1F1[−n ; 1 + α ; x].
It is known (see, e.g., [15, Section 48]) that there are 3p linearly independent contiguous function relations forpFp. Using the three contiguous relations for 1F1 with the aid of (45), the three mixed recurrence relations for L
(α) n (x) are established (see [15, p. 203, Eqs. (8), (9) and (10)]), for example,
(46) L(α)n (x) = L(α)n−1(x) + L(α−1)n (x).
Similarly, 3p different recurrence relations for L(α)n,p(x) may be obtained from the 3p contiguous relations for pFp. Unfortunately, no recurrence relations for L(α)n,p(x) (p ≥ 2) can be derived from the 3p contiguous relations forpFp. Indeed, if 1 is added or subtracted at one of the numerator or denominator parameters in the right member of (44), the other parameters cannot be expressed in the same fashion as in (44) whenever p ≥ 2. The generalized Laguerre polynomials introduced here and their properties and formulas presented are hoped to be potentially useful.
Acknowledgements. The author express their sincere gratitude to Professor Junesang Choi and Muhammad Zafar Iqbal for useful discussions and invalu-able advice. And also grateful to the editor and the referee for carefully reading the manuscript and for valuable comments and suggestions which greatly im-proved this paper. I have dedicated this paper to my parents whose constant encouragement has enabled me to do this work.
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Asad Ali
Department of Mathematics and Statistics University of Agriculture Faisalabad Faisalabad 38000, Pakistan
