On an Inequality for Legendre Polynomials

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Article On an Inequality for Legendre Polynomials

Florin Sofonea *,† and Ioan ¸Tincu †

Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Str. Dr. I. Ratiu, No.5-7, RO-550012 Sibiu, Romania; [email protected] * Correspondence: ﬂ[email protected] † These authors contributed equally to this work.

Received: 17 September 2020; Accepted: 13 November 2020; Published: 17 November 2020

Abstract: This paper is concerned with the orthogonal polynomials. Upper and lower bounds of Legendre polynomials are obtained. Furthermore, entropies associated with discrete probability distributions is a topic considered in this paper. Bounds of the entropies which improve some previously known results are obtained in terms of inequalities. In order to illustrate the results obtained in this paper and to compare them with other results from the literature some graphs are provided.

Keywords: Legendre; Chebyshev; Gegenbauer; hypergeometric representation

1. Introduction The classical orthogonal polynomials play an important role in applications of mathematical analysis, spectral method with applications in ﬂuid dynamics and other areas of interest. In the recent years many theoretical and numerical studies about Jacobi polynomials were given. Inequalities for Jacobi polynomials using entropic and information inequalities were obtained in [1]. Bernstein type inequalities for Jacobi polynomials and their applications in many research topics in mathematics were considered in [2]. A very recent conjecture (see [3]) which asserts that the sum of the squared Bernstein polynomials is a convex function in [0, 1] was validated using properties of Jacobi polynomials.This conjecture aroused great interest, so that new proofs of it were given (see [4,5]). The main objective of this paper is to obtain upper and lower bounds of Legendre polynomials and to apply these results in order to give lower and upper bounds for entropies. Usually the entropies are described by complicated expressions, then it is useful to establish some bounds of them. The concept of entropy is a measure of uncertainty of a random variable and was introduced by C.E. Shannon in [6]. Later, A. Rényi [7] introduced a parametric family of information measures, that includes Shannon entropy as a special case. The classical Shannon entropy was given using discrete probability distributions. This concept was extended to the continuous case involving the continuous probability distributions. For more details about this topic, the reader is referred to the recent papers [8–12]. This paper is devoted to the entropy associated with discrete probability distribution. (α,β) α β The Jacobi polynomials Pn are orthogonal polynomials with respect to the weight (1 − x) (1 + x) , α > −1, β > −1 on the interval [−1, 1]. Their representation by hypergeometric function is given as follows (see [13,14]) (α,β) (α + 1) 1 − x P (x) = n F −n, n + α + β + 1; α + 1; , (1) n n! 2 1 2

Mathematics 2020, 8, 2044; doi:10.3390/math8112044 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 2044 2 of 11 where ∞ (a) (b) zk F (a, b; c; z) = k k · , |z| < 1, 2 1 ∑ ( ) k=0 c k k! and (a)k = a(a + 1) ... (a + k − 1), k ≥ 1, (a)0 = 1 is the shifted factorial. (α,β) The normalization of Pn is effected by

(α,β) Γ(n + α + 1) P (1) = . n n!Γ(α + 1)

(α,β) Denote by Rn the polynomials:

(α,β) (α,β) P (x) R (x) = n . (2) n (α,β) Pn (1) (α,β) If α = β = λ, Jacobi’s polynomials Pn (x) are called Gegenbauer polynomials (ultraspherical polynomial). The usual notation and normalization for ultraspherical polynomial is the following (see [15])

( ) Γ(α + 1) Γ(n + 2α + 1) ( ) C λ (x) = P α,α (x) n Γ(2α + 1) Γ(n + α + 1) n + 1 Γ λ 2 Γ(n + 2λ) (λ− 1 ,λ− 1 ) 1 = P 2 2 (x), α = λ − . Γ(2λ) 1 n 2 Γ(n + λ + 2 )

Note that ( ) (2λ) (λ− 1 ,λ− 1 ) (2λ) 1 1 − x C λ (x) = n R 2 2 (x) = n F −n, n + 2λ; λ + ; . (3) n n! n n! 2 1 2 2

Some important special cases are the Chebyshev polynomial of the ﬁrst kind

− 1 ,− 1 ( 2 2 ) 1 1 Pn (x) (− 2 ,− 2 ) Tn(x) = 1 1 = Rn (x), (4) (− 2 ,− 2 ) Pn (1) the Chebyshev polynomial of the second kind

1 , 1 ( 2 2 ) 1 1 (1) Pn (x) ( 2 , 2 ) Un(x) = Cn (x) = (n + 1) 1 1 = (n + 1)Rn (x), (5) ( 2 , 2 ) Pn (1) and the Legendre polynomial 1 ( 2 ) (0,0) Pn(x) = Cn (x) = Pn (x). (6)

2. Preliminary Results 1 2k 2n − 2k + 2 Theorem 1. Let Bn,k := · · , k, n ∈ N, k ≤ n. Then, the inequalities 22n+1 k n − k + 1

αn < Bn,k < βn, (7) Mathematics 2020, 8, 2044 3 of 11 hold for all n ∈ N, k ∈ N \{0}, k ≤ n, where

1 4 αn := · , π + + 1 + 1 n 1 2 8(n+1) √ 1 4 4 46 β := · . n π q √ 1 + (8 + 46)(n + 1)

Proof. In order to prove the inequalities (7) we use Tóth’s result (see [16]):

1 1 q < Ωn < q , (8) 1 1 1 1 π(n + 4 + 32n ) π(n + 4 + 46n ) ! 1 · 3 ··· (2n − 1) 1 2n ∗ where Ωn = = for n ∈ N . 2 · 4 ··· (2n) 4n n From (8), it follows that

2 1 · < B (9) π r h i n,k ( + 1 + 1 ) − + + 1 + 1 k 4 32k n k 1 4 32(n−k+1) 2 1 < · . π r h i ( + 1 + 1 ) − + + 1 + 1 k 4 46k n k 1 4 46(n−k+1)

1 1 Let α ∈ [ 46 , 32 ], and consider f : D → R a function deﬁned by

1 α 1 α f (x, y) = x + + y + + , 4 x 4 y where D = {(x, y) ∈ R2 x > 0, y > 0, x + y = n + 1}. Further, we determine the extreme points of function f . We consider F(x, y) = f (x, y) + λ(x + y − n − 1). From the system

 0  Fx = 0  0 Fy = 0  0 Fλ = x + y − n − 1 = 0, we get  4x2y2 − 4αxy − 4α(x2 + y2) − 4αxy − α(x + y) − 4α2  (y − x) · = 0, 4x2y2  (y − x){4x2y2 − α[4(n + 1)2 + n + 1 + 4α]} = 0, Solving the sistem we obtain the following stationary points: √ √ ! √ √ ! n + 1 n + 1 n + 1 − ∆0 n + 1 + ∆0 n + 1 + ∆0 n + 1 − ∆0 A , , B , , C , , 2 2 2 2 2 2

p where ∆0 = (n + 1)2 − 2 α[4(n + 1)2 + (n + 1) + 4α]. Mathematics 2020, 8, 2044 4 of 11

We have,

2 00 2 00 2 00 00 00 2 d F(x, y) = Fx2 (x, y)dx + 2Fxy(x, y)dxdy + Fy2 (x, y)dy = [Fx2 (x, y) − 2Fxy + Fy2 (x, y)]dx .

Since d2F(A) < 0, it follows that A is a maximum point for f . For (x, y) ∈ {B, C}, we have

α (n + 1)4 − 4α[4(n + 1)2 + (n + 1) + 4α] d2F(x, y) = [4(n + 1)2 + (n + 1) + 4α] dx2 > 0. 2x3y3 (n + 1)2 + 4xy

Then f attains its minimum value at the points B and C We compute √ ! √ ! n + 1 − ∆0 1 2α n + 1 + ∆0 1 2α f (B) = + + √ + + √ 2 4 n + 1 − ∆0 2 4 n + 1 + ∆0 √ q 4n + 5 − 32α = α · 4(n + 1)2 + (n + 1) + 4α + . 16 We observe that r 1 2 n + 1 1 1 n + 1 f (C) = f (B) > √ 4(n + 1)2 + (n + 1) + + > √ 2n + 2 + + , 46 23 4 46 4 4 √ 1 + (8 + 46)(n + 1) 1 1 4α 2 f (B) > √ , f (A) = n + 1 + + . 4 46 4 2 n + 1 It follows f (B) ≤ f (x, y) ≤ f (A), 1 1 1 p ≤ p ≤ p , f (A) f (x, y) f (B) √ 2 1 2 4 46 ≤ ≤ . + + 1 + 4α p ( ) q √ n 1 2 n+1 f x, y 1 + (8 + 46)(n + 1)

1 1 For α ∈ , and k ∈ N \{0}, we obtain (7). 32 46

Remark 1. The bounds of Bn,k for n ∈ N and k = 0 were obtained [16], as follows     1 2n  2 2  B − = ∈  ,  . (10) n 1,0 22n−1 n  s s   1 1 1 1  π n + 4 + 32n π n + 4 + 46n

From Theorem1 and (10), we get αn < Bn,k < γn, (11) 2 for all k = 0, 1, . . . , n, where γ := . n r + + 1 + 1 π n 1 4 46(n+1) Mathematics 2020, 8, 2044 5 of 11

3. Bounds for Legendre Polynomials Lemma 1. Let b n c " r !#2 2 1 + x Sn(x) := ∑ Un−2k , x ∈ (−1, 1), k=0 2 where bxc is integer part of positive real number x. The following relation holds

1 S (x) = [n + 2 − U + (x)] . (12) n 2(1 − x) n 1

(α,β) Proof. From (1) and (2) the polynomials Rn admite the following representation by hypergeometric function (α,β) 1 − x R = F −n, n + α + β + 1; α + 1; . (13) n 2 1 2

From [14] (eq. 4.7.30, p. 83) we have (λ− 1 ,λ− 1 ) n n 1 R 2 2 (x) = F − , + λ; λ + ; 1 − x2 . (14) n 2 1 2 2 2

Therefore, ( 1 , 1 ) 3 R 2 2 (x) = F −n, n + 1; ; 1 − x2 , 2n 2 1 2 ( 1 , 1 ) 1 3 3 R 2 2 (x) = F −n − , n + ; ; 1 − x2 . 2n+1 2 1 2 2 2 Using the below formula (see [17] ((15.3.3), p. 559))

c−a−b 2F1(a, b; c; z) = (1 − z) 2F1(c − a, c − b; c; z), |z| < 1, we obtain ( 1 , 1 ) 3 R 2 2 (x) = x F (n + 2, −n; ; 1 − x2). 2m+1 2 1 2 Then, r ! r ( 1 , 1 ) 1 + x 1 + x ( 1 , 1 ) R 2 2 = R 2 2 (x). (15) 2n+1 2 2 n

The link between Chebyshev polynomial of the ﬁrst and second kind are presented in the following two relations (see [14] (Theorem 4.1, p. 59))

2 T2n(x) = Tn(2x − 1), (16) 2 2 2(1 − x )Un(x) = 1 − T2n+2(x). (17) Mathematics 2020, 8, 2044 6 of 11

Using relations (15) and (17) we get

n " r !#2 ( 1 , 1 ) 1 + x ( ) = ( − + )2 2 2 S2n+1 x ∑ 2n 2k 2 R2n−2k+1 k=0 2 n 2 n x + 1 ( 1 , 1 ) = ( − + )2 2 2 ( ) = ( + ) ( ) 2 ∑ 2n 2k 2 Rn−k x ∑ 2 x 1 [Un−k x ] k=0 2 k=0 n 1 n 1 − T (x) = [1 − T (x)] = 2k+2 . ∑ − 2n−2k+2 ∑ − k=0 1 x k=0 1 x

From relations (16) and (17) it follows

" r !#2 " r !# n 1 + x n 1 1 + x S (x) = U = 1 − T 2n ∑ 2n−2k ∑ − 4n−4k+2 k=0 2 k=0 1 x 2 n 1 − T (x) n 1 − T (x) = 2n−2k+1 = 2k+1 . ∑ − ∑ − k=0 1 x k=0 1 x

Using the equalities

n 1 n 1 ∑ T2k+1(x) = U2n+1(x), ∑ T2k(x) = [1 + U2n(x)] , k=0 2 k=0 2 we obtain, after a straight forward computation: " # n 1 − T (x) 1 n S (x) = 2k+2 = n + 1 − T (x) 2n+1 ∑ − − ∑ 2k+2 k=0 1 x 1 x k=0 " # " # 1 n+1 1 n+1 = n + 1 − T (x) = n + 2 − T (x) − ∑ 2k − ∑ 2k 1 x k=1 1 x k=0 1 = [2n + 3 − U + (x)] , 2(1 − x) 2n 2

" # n 1 − T (x) 1 n 1 S (x) = 2k+1 = n + 1 − T (x) = [2n + 2 − U (x)] . 2n ∑ − − ∑ 2k+1 ( − ) 2n+1 k=0 1 x 1 x k=0 2 1 x

Therefore 1 S (x) = [n + 2 − U + (x)]. n 2(1 − x) n 1

Theorem 2. Let Pn be the Legendre polynomial of degree n, n ∈ N. Then, for x ∈ (−1, 1)

2 1 [n + 1 − Un(x)] < 1 − Pn(x) < [n + 1 − Un(x)] , 1 1 r π n + 2 + 8n 1 1 π n + 4 + 46n where Un is the Chebyshev polynomial of the second kind. Mathematics 2020, 8, 2044 7 of 11

Proof. Using the relation (see [18] (p. 713))

n n [ ] 1 α+2 α+3 " r !#2 ( ) 2 ( ) ( ) − ( ) − (n − 2k)! ( α+1 ) 1 + x P α,0 (x) = 2 k 2 n k 2 n 2k C 2 , ∑ k ∑ α+3 α+1 n−2k 2 k=0 k=0 k!( 2 )n−k( 2 )n−2k(α + 1)n−2k we get

n n b c 1 3 " r !#2 ( ) 2 ( ) ( ) − ( ) 1 + x P 1,0 (x) = 2 k 2 n k C 1 ∑ k ∑ ( ) n−2k k=0 k=0 k! 2 n−k 2 b n c " r !#2 2 1 + x = (1) ∑ Bn,k Cn−2k . (18) k=0 2

(α,β) The Jacobi polynomials Pn , n ∈ N, α > −1, β > −1 verify the following equality (see [14] ((4.5.4), p. 71)): (α+1,β) 2 1 h (α,β) (α,β) i P (x) = · · (n + α + 1)P (x) − (n + 1)P (x) . (19) n 2n + α + β + 2 1 − x n n+1 From relations (5), (18) and (19) we obtain

b n c " r !#2 1 − P + (x) 2 1 + x n 1 = B U . − ∑ n,k n−2k 1 x k=0 2

Using relation (11), we get

b n c " r !#2 b n c " r !#2 b n c " r !#2 2 1 + x 2 1 + x 2 1 + x αn ∑ Un−2k < ∑ Bn,k Un−2k < γn ∑ Un−2k . k=0 2 k=0 2 k=0 2

Therefore, 1 − P + (x) α S < n 1 < γ S . n n 1 − x n n Consequently,

α − γ − n 1 [n + 1 − U (x)] < 1 − P (x) < n 1 [n + 1 − U (x)] , (20) 2 n n 2 n and the Theorem is proved.

2 [n + 1 − Un(x)] [n + 1 − Un(x)] Example 1. Let un(x) := , vn(x) := . In Figure1 are given the graphics 1 1 r π n + 2 + 8n 1 1 π n + 4 + 46n of the polynomial 1 − Pn(x) and the lower and the upper bounds of it obtained in Theorem2. Mathematics 2020, 8, 2044 8 of 11

Figure 1. Graphs of un, vn and 1 − Pn.

4. Bounds for Information Potentials

Let p(x) = (pk(x))k≥0 be a parameterized discrete probability distribution. The associated information potential is deﬁned by ∞ 2 S(x) = ∑ pk(x), x ∈ T, where I is a real interval. k=0 The Rényi entropy of order 2 and the Tsallis entropy of order 2 associated with p(t, x) can be expressed in terms of the information potential S(x) as follows (see [18] (pp. 20–21)):

R(x) := − log S(x); T(x) = 1 − S(x).

[−1] [0] In the following, we consider the binomial distribution pn,k , the Poisson distribution pn,k and the [1] negative binomial distribution pn,k: [− ] n p 1 (x) = xk(1 − x)n−k, x ∈ [0, 1], n,k k k [ ] (nx) p 0 (x) = e−nx , x ≥ 0, n,k k! [ ] n + k − 1 p 1 (x) = xk(1 + x)−n−k, x ≥ 0. n,k k

The associated information potentials (index of coincidence) of this distributions are deﬁned as follows:

n 2 n k n−k Sn,−1(x) := ∑ x (1 − x) , x ∈ [0, 1], (21) k=0 k ∞ (nx)2k ( ) = −2nx ≥ Sn,0 x : e ∑ 2 , x 0, (22) k=0 (k!) ∞ 2 n + k − 1 k −n−k Sn,1(x) := ∑ x (1 + x) , x ≥ 0. (23) k=0 k Mathematics 2020, 8, 2044 9 of 11

From [10] (56) and [12] (21) we know that

n 1 2k Sn,−1(x) = ∑ Bn−1,k(1 − 2x) , 2 k=0 n−1 1 −2k−1 Sn+1,1(x) = ∑ Bn−1,k(1 + 2x) . 2 k=0

The following lower and upper bounds for the information potential associated to binomial distribution was obtained in [11]:

2n 1 − n− [1 + (n − 3)x(1 − x)] 3 ≤ Sn,−1(x) ≤ p . (24) 1 + 4(n − 1)x(1 − x)

Using Remark1 can be obtained new bounds for the information potential Sn,−1 and Sn,1 as follows:

Theorem 3. The following inequalities are satisﬁed

1 1 − (1 − 2x)2n+2 1 1 − (1 − 2x)2n+2 < Sn,−1(x) < , x ∈ [0, 1], (25) 1 1 2x(1 − x) r 4x(1 − x) π n + 2 + 8n 1 1 π n + 4 + 46n

1 1+2x 1 1− < Sn+1,1(x) 1 1 2x(1+x) (1 + 2x)2n π n+ 2 + 8n 1 1+2x 1 < − > r 1 2n , x 0. 1 1 4x(1+x) (1+2x) π n + 4 + 46n

Remark 2. Let n = 10. In Figure2 we give a graphical representation of the lower bound of Sn,−1(x) from (24) and (25), respectively. Denote these bounds with LS(24) and LS(25), respectively. In Figure3 we give a graphical representation of the upper bound of Sn,−1(x) from (24) and (25), respectively. Denote these bounds with RS(24) and RS(25), respectively. Remark that on certain intervals the results obtained in Theorem3 improve the result from [ 11].

Figure 2. Graphs of lower bounds of Sn,−1. Mathematics 2020, 8, 2044 10 of 11

Figure 3. Graphs of upper bounds of Sn,−1.

5. Conclusions This paper is devoted to the orthogonal polynomials. Bounds of Legendre polynomials are obtained in terms of inequalities. A more general result in regard with the estimate of the coefﬁcients Bn,k is obtained and used in order to give bounds of information potentials associated of the binomial distribution and the negative binomial distribution. The bounds of the entropies are useful especially when they are described by complicated expressions. The results obtained in this paper improve some results from the literature. Motivated by many applications of entropies in secure data transmission, speech coding, cryptography, algorithmic complexity theory, ﬁnding such bounds for the entropies will be a topic for future work.

Author Contributions: These authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript. Funding: Project financed by Lucian Blaga University of Sibiu & Hasso Plattner Foundation research grants LBUS-IRG-2019-05. Acknowledgments: The authors are very greateful to the reviewers for their valuable comments and suggestions. Conflicts of Interest: The authors declare no conflict of interest.

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