Skip to main content

Advertisement

Two inequalities for the associated Pollaczek polynomials

  • 120 Accesses

Abstract

In this paper, we prove two concise inequalities for the associated Pollaczek polynomials. The first one is obtained by using Askey’s theorem (SIAM J. Math. Anal. 2:340–346, 1971) on orthogonal expansions with positive coefficients. The second one is proved by using a triple integral representation due to the authors (Integral Transforms Spec. Funct. 30:893–919, 2019). In the concluding section, we briefly point out some useful variations and known cases of our inequalities.

Introduction and main result

The associated Pollaczek polynomials \(P_{n}^{\lambda} (\cos\theta; a,b,c )\) can be defined by (see, e.g., [13, 14] and [17])

$$\begin{aligned} \sum_{n=0}^{\infty}P_{n}^{\lambda} (\cos\theta; a,b,c )t^{n} &= \bigl(1-t\mathrm{e}^{-\mathrm{i}\theta} \bigr)^{-\lambda-\mathrm {i}\varPhi} \bigl(1-t\mathrm{e}^{\mathrm{i}\theta} \bigr)^{-\lambda+\mathrm {i}\varPhi} \\ &\quad\times {F_{1} \bigl[c, 1-\lambda-\mathrm{i}\varPhi, 1-\lambda+\mathrm{i}\varPhi; c+1; t\mathrm{e}^{-\mathrm{i}\theta}, t \mathrm{e}^{\mathrm{i}\theta} \bigr]}, \end{aligned}$$
(1.1)

where \(|t|<1\),

$$ \varPhi\equiv\varPhi (\theta ):=\frac{a\cos\theta+b}{\sin\theta}, $$
(1.2)

and \(F_{1}\) is the Appell hypergeometric function defined by ([18, p. 53, Eq. (4)])

$$ F_{1} [\alpha,\beta_{1},\beta_{2};\gamma;x,y ] = \sum_{m,n=0}^{\infty}\frac{ (\alpha )_{m+n} (\beta_{1} )_{m} (\beta_{2} )_{n}}{ (\gamma )_{m+n}} \frac{x^{m}}{m!}\frac {y^{n}}{n!}, \quad\max\bigl\{ \vert x \vert , \vert y \vert \bigr\} < 1. $$

Here the Pochhammer symbol \((\lambda )_{n}\) is defined (for \(\lambda\in\mathbb{C}\)) by

$${ (\lambda )_{n}:= \textstyle\begin{cases} 1 & (n=0 ), \\ \lambda (\lambda+1 )\cdots (\lambda+n-1 ) & (n\in \mathbb{N}:=\{1,2,\ldots\} ). \end{cases}\displaystyle } $$

When \(c=0\), we may obtain from (1.1) the generating function of the Pollaczek polynomials \(P_{n}^{\lambda} (\cos\theta; a,b )\):

$$ \sum_{n=0}^{\infty}P_{n}^{\lambda} (\cos\theta; a,b )t^{n} = \bigl(1-t\mathrm{e}^{-\mathrm{i}\theta} \bigr)^{-\lambda-\mathrm {i}\varPhi} \bigl(1-t\mathrm{e}^{\mathrm{i}\theta} \bigr)^{-\lambda+\mathrm {i}\varPhi}. $$
(1.3)

Conventionally, the polynomials obtained by setting \(\lambda=\frac {1}{2}\) in \(P_{n}^{\lambda} (\cos\theta; a,b )\) are simply denoted by \(P_{n} (\cos\theta; a,b )\). For classical results on Pollaczek polynomials, we refer the interested reader to [6, 7, 10], and [16]. Furthermore, readers are encouraged to see [13, 14] and [9] for the latest development in this area.

Despite some important results, we still know little about the associated Pollaczek polynomials, especially about the inequalities satisfied by this class of polynomials. In this paper, our aim is to establish two interesting inequalities for the associated Pollaczek polynomials.

The first inequality is given by the following:

Theorem 1.1

For \(a,b,c\geq0\), \(\lambda>0\), and \(x\in[-1,1]\), we have

$$ \bigl\vert P_{n}^{\lambda} (x;a,b,c ) \bigr\vert \leq P_{n}^{\lambda} (1;a,b,c ). $$
(1.4)

As we will show in the following sections, Theorem 1.1 is obtained by using an important result from Askey’s work on orthogonal expansions with positive coefficients. Theorem 1.2 is obtained by using a completely different method.

Theorem 1.2

For \(\theta\in (0,\pi )\), \(c>0\), \(\lambda>0\), and \(c+2\lambda >1\), we have

$$ \bigl\vert P_{n}^{\lambda} (\cos\theta;a,b,c ) \bigr\vert \leq\frac{[\varGamma (\lambda )]^{2}}{ \vert \varGamma (\lambda+\mathrm{i}\varPhi ) \vert ^{2}} \frac{ (c+2\lambda )_{n} (2c+2\lambda )_{n}}{ (c+1 )_{n} n!} (1+2\sin\theta )^{n}, $$
(1.5)

whereΦis given by (1.2).

We denote by \(P_{n} (\cos\theta;a,b,c )\) the polynomials obtained by letting \(\lambda=\frac{1}{2}\) in \(P_{n}^{\lambda} (\cos \theta; a,b,c )\). Then, using Theorem 1.2, we have

$$ \operatorname{sech} (\pi\varPhi ) \bigl\vert P_{n} (\cos\theta;a,b,c ) \bigr\vert \leq\frac{ (2c+1 )_{n}}{n!} (1+2\sin\theta )^{n}, $$

where we have used the relation that \(\vert \varGamma (\tfrac {1}{2}+\mathrm{i}y ) \vert ^{2}=\pi\operatorname{sech} (\pi y )\) (see [16, p. 137, Eq. (5.4.4)]).

Key lemmas

In this section, we present some useful lemmas used in our proofs. The first result due to Askey [3] gives a sufficient condition for writing a set of orthogonal polynomials as a linear combination of a second set of orthogonal polynomials with nonnegative coefficients. We will use it in the proof of Theorem 1.1.

Let \(p_{n} (x )\) be defined by

$$ xp_{n} (x )=p_{n+1} (x )+\alpha_{n} p_{n} (x )+\beta_{n} p_{n-1} (x ),\quad n=0,1,2,\ldots, $$
(2.1)

where \(p_{-1} (x )=0\), \(p_{0} (x )=1\), \(\alpha_{n-1}\) real, \(\beta_{n}>0\), \(n=1,2,\dots \), and the polynomials are normalized to be monic (i.e., the leading coefficients of the polynomials are one).

Similarly, let \(p_{n} (x )\) be defined by

$$ xq_{n} (x )=q_{n+1} (x )+\gamma_{n} q_{n} (x )+\delta_{n} q_{n-1} (x ),\quad n=0,1,2,\ldots, $$
(2.2)

where \(q_{-1} (x )=0\), \(q_{0} (x )=1\), \(\gamma_{n-1}\) real, \(\delta_{n}>0\), \(n=1,2,\dots \). Then Askey’s result can be stated as follows.

Lemma 2.1

([3, p. 341])

Let \(p_{n} (x )\)and \(q_{n} (x )\)be defined by (2.1) and (2.2) and set

$$q_{n} (x )=\sum_{k=0}^{n}a (k,n ) p_{k} (x ). $$

Then \(a (k,n )\geq0\)if

$$\begin{aligned}& \alpha_{k}\geq\gamma_{n}, \quad k=0,1,\ldots, n, n=0,1, \ldots, \end{aligned}$$
(2.3)
$$\begin{aligned}& \beta_{k}\geq\delta_{n}, \quad k=0,1,\ldots, n, n=0,1, \ldots. \end{aligned}$$
(2.4)

To use this lemma, we require three-term recurrence relations for the normalized (monic) associated Pollaczek polynomials and some of their particular cases. Let \(\hat{P}_{n}^{\lambda} (x; a,b,c )\) denote the normalized associated Pollaczek polynomials. We have ([6, p. 185])

$$ \hat{P}_{n}^{\lambda} (x;a,b,c )=\frac{ (c+1 )_{n}}{2^{n} (\lambda+a+c )_{n}} P_{n}^{\lambda} (x;a,b,c ), $$

from which we have the following three-term recurrence relation ([6, p. 185, Eq. (5.9)]):

$$\begin{aligned} x {\hat{P}_{n}^{\lambda}(x;a,b,c )}& =\hat{P}_{n+1}^{\lambda}(x;a,b,c ) -\frac{b}{n+\lambda+a+c} \tilde{P}_{n}^{\lambda}(x;a,b,c ) \\ &\quad +\frac{ (n+c ) (n+2\lambda+c-1 )}{4 (n+\lambda +a+c ) (n+\lambda+a+c-1 )} \hat{P}_{n-1}^{\lambda}(x;a,b,c ). \end{aligned}$$
(2.5)

When \(a=b=0\), (2.5) reduces to the three-term recurrence relation of the normalized associated ultraspherical (Gegenbauer) polynomials \(\hat{C}_{n}^{\lambda}(x,c )\) given by

$$ x {\hat{C}_{n}^{\lambda}(x,c )} =\hat{C}_{n+1}^{\lambda}(x,c ) +\frac{ (n+c ) (n+2\lambda+c-1 )}{4 (n+\lambda +c ) (n+\lambda+c-1 )} \hat{C}_{n-1}^{\lambda}(x,c ). $$
(2.6)

It was also proved in [3, p. 345] that

$$ \bigl\vert \hat{C}_{n}^{\lambda} (x,c ) \bigr\vert \leq\hat {C}_{n}^{\lambda} (1,c ),\quad x\in[-1,1], \lambda>0, c \geq0. $$
(2.7)

Following Carlson [5, p. 52, Def. 3.11-1], we define the Euler measure \(\mathfrak{m}_{\alpha}\) on \(\mathbb{R}_{+}\) by

$$ \mathrm{d}\mathfrak{m}_{\alpha} (u ):=\frac{1}{\varGamma (\alpha )} u^{\alpha-1} \mathrm{e}^{-u}\,\mathrm{d}u\quad (\alpha>0 ). $$

It is not difficult to verify that \(\mathfrak{m}_{\alpha} (\mathbb {R}_{+} )=1\). For \(\Re (\alpha )>0\) and \(\Re (\beta )>0\), we define

$$ \mathrm{d}\mu_{\alpha, \beta} (t ):= \frac{\varGamma (\alpha+\beta )}{\varGamma (\alpha )\varGamma (\beta )} t^{\alpha-1} (1-t )^{\beta-1}\,\mathrm{d}t, $$
(2.8)

which is a particular case of the Dirichlet measure (see [5, p. 59]) and satisfies \({\mu_{\alpha,\beta} ([0,1] )=1}\). Throughout this paper, we let

$$ \mathrm{d}\varPi (t,u_{1},u_{2} ) := \mathrm{d}\mu_{c,\lambda+\mathrm{i}\varPhi} (t ) \,\mathrm{d}\mathfrak{m}_{c+2\lambda-1} (u_{1} ) \,\mathrm{d}\mathfrak{m}_{1} (u_{2} ). $$
(2.9)

The following Lemma 2.2 is proved in [13] and is essential in the proof of Theorem 1.2.

Lemma 2.2

For \(c>0\), \(\lambda>0\), \(c+2\lambda>1\), we have

Pnλ(cosθ;a,b,c)=(c+2λ)nn!einθ×(0,1)×R+22F2[n,c+λ+iΦc+1,c+2λ;tu1+u2(1e2iθ)]dΠ(t,u1,u2),
(2.10)

where \({}_{2}F_{2}\)is the generalized hypergeometric function defined by

F22[a1,a2b1,b2;z]:=n=0(a1)n(a2)n(b1)n(b2)nznn!(zC).
(2.11)

In addition to this result, the proof of Theorem 1.2 heavily relies on the properties of hypergeometric functions and Laguerre polynomials. The following lemma giving an Eulerian-type integral representation for \({}_{2}F_{2}\)-function enables us to handle appropriately the \({}_{2}F_{2}\)-function occurring in (2.10).

Lemma 2.3

([16, p. 408, Eq. (16.5.2)])

For \(\Re (b_{2} )>\Re (a_{2} )>0\), we have

F22[a1,a2b1,b2;z]=Γ(b2)Γ(a2)Γ(b2a2)01ta21(1t)b2a211F1[a1b1;zt]dt,
(2.12)

where \({}_{2}F_{2}\)is defined by (2.11), and \({}_{1}F_{1}\)is the confluent hypergeometric function (see, e.g., [18, p. 36, Eq. (3)]).

Also, we will frequently use the following version of the Chu–Vandermonde identity (see, e.g., [18, p. 31]).

Lemma 2.4

$$ (x+y )_{n}=\sum_{\ell=0}^{n} \binom{n}{\ell} (x )_{\ell } (y )_{n-\ell}. $$
(2.13)

For the Laguerre polynomials defined by [16, p. 443, Eq. (18.5.12)]

Ln(α)(x):=(α+1)nn!1F1[nα+1;x],
(2.14)

Eqs. (2.15) and (2.16) given further by Lemma 2.5 are known as the addition and multiplication theorems for the Laguerre polynomials, respectively, and would be required in the proof of our inequality (1.5).

Lemma 2.5

(see [16, p. 460, Eq. (18.18.12)] and [16, p. 461, Eq. (18.18.38)])

$$\begin{aligned}& L_{n}^{(\alpha+\beta+1)} (x+y )=\sum _{\ell=0}^{n}L_{\ell }^{(\alpha)} (x )L_{n-\ell}^{(\beta)} (y ), \end{aligned}$$
(2.15)
$$\begin{aligned}& \frac{L_{n}^{(\alpha)} (\lambda x )}{L_{n}^{(\alpha)} (0 )}=\sum_{\ell=0}^{n} \binom{n}{\ell}\lambda^{\ell}(1-\lambda )^{n-\ell} \frac{L_{\ell}^{(\alpha)} (x )}{L_{\ell }^{(\alpha)} (0 )}. \end{aligned}$$
(2.16)

The Laguerre polynomials satisfy the following well-known important inequality (see [1, p. 786, Eq. (22.14.13)]):

$$ \bigl\vert L_{n}^{(\alpha)} (x ) \bigr\vert \leq \frac{ (\alpha+1 )_{n}}{n!}\mathrm{e}^{x/2}\quad (\alpha\geq0, x\geq0 ), $$
(2.17)

In 1997, Love [12] published several inequalities for the Laguerre polynomials \(L_{n}^{(\alpha)} (x )\) (with complex α) and for the Laguerre functions \(L_{\nu}^{(\mu)} (x )\) (with complex μ and ν).

Although inequality (2.17) is quite elegant, the involved exponential factor \(\mathrm{e}^{x/2}\) can make an integral to be divergent. The same exponential factor occurs in Love’s generalization of (2.17) (see [12, p. 295, Theorem 1]). So we need a particular bound for Laguerre polynomials. To remedy such a situation, we denote by \(\sigma_{n}^{(\alpha)}\) the Cesáro mean defined by

$$\sigma_{n}^{(\alpha)} \Biggl(\sum_{n=1}^{\infty}a_{n} \Biggr) =\frac{n!}{ (\alpha+1 )_{n}} \sum_{k=0}^{n} \frac{ (\alpha+1 )_{n-k}}{ (n-k )!}a_{k}. $$

Lemma 2.6

([11, p. 532, Eq. (10)])

For \(\alpha\geq-\frac{1}{2}\), \(x\geq0\), and \(n=0,1,\dots\), we have

$$ \bigl\vert L_{n}^{(\alpha)} (x ) \bigr\vert \leq \frac{ (\alpha+1 )_{n}}{n!}\sigma_{n}^{(\alpha)} \bigl(\mathrm{e}^{x} \bigr), $$
(2.18)

where

$$ \sigma_{n}^{(\alpha)} \bigl(\mathrm{e}^{x} \bigr) = \frac{n!}{ (\alpha+1 )_{n}} \sum_{k=0}^{n} \frac{ (\alpha+1 )_{n-k}}{ (n-k )!}\frac{x^{k}}{k!}. $$

Estimate (2.18) is better than (2.17) for large x.

Proof of Theorem 1.1

In view of the range of the parameter λ, the proof is divided into two parts. We first prove inequality (1.4) for \(0<\lambda<1\). We then consider the case \(\lambda\geq1\).

When \(0<\lambda<1\), our aim is to show that the normalized associated Pollaczek polynomials can be written as linear combinations of some normalized associated ultraspherical polynomials with nonnegative coefficients. More precisely, we want to prove that

$$ \hat{P}_{n}^{\lambda} (x;a,b,c ) =\sum _{k=0}^{n}a (k,n )\hat{C}_{n}^{\lambda} (x,c ),\quad \text{where $a (k,n )\geq0$.} $$
(3.1)

By comparing (2.1) with (2.6) we have

$$\alpha_{n}=0 \quad\text{and}\quad \beta_{n}=\frac{ (n+c ) (n+2\lambda+c-1 )}{4 (n+\lambda+c ) (n+\lambda+c-1 )}, $$

where \(\beta_{n}>0\) for \(n=1,2,\dots \). Similarly, comparing (2.2) with (2.5) gives

$$\gamma_{n}=-\frac{b}{n+\lambda+a+c} \quad\text{and}\quad \delta_{n}= \frac{ (n+c ) (n+2\lambda+c-1 )}{4 (n+\lambda+a+c ) (n+\lambda+a+c-1 )} , $$

where \(\delta_{n}>0\) for \(n=1,2,\dots \). Obviously, we have \(\beta_{n}\geq \delta_{n}\) because \(a\geq0\). We also have

$$\alpha_{k}\geq\gamma_{n},\quad k=0,1,\ldots, n, n=0,1,\ldots, $$

since \(a,b,c\geq0\) and \(0<\lambda<1\). So condition (2.3) of Lemma 2.1 is satisfied.

To prove that \(\beta_{k}\geq\delta_{n}\), \(k=0,1,\ldots, n\), \(n=0,1,\dots \), we need to show that

$$ f (x ):=\frac{x (x+2\lambda-1 )}{4 (x+\lambda ) (x+\lambda-1 )} $$

is a decreasing function on \((1-\lambda,\infty )\). By taking the logarithmic derivative with respect to x, we obtain

$$ \frac{\mathrm{d}}{\mathrm{d}x}\log f (x ) =\frac{1}{x}+\frac{1}{x+2\lambda-1}- \frac{1}{x+\lambda}-\frac {1}{x+\lambda-1}. $$

The condition under which \(\frac{\mathrm{d}}{\mathrm{d}x}\log f (x )<0\) is obtained by observing that

$$\begin{aligned} \frac{1}{x}+\frac{1}{x+2\lambda-1}< \frac{1}{x+\lambda}+\frac {1}{x+\lambda-1} &\quad \Leftrightarrow \quad(x+\lambda ) (x+\lambda-1 )< x (x+2\lambda-1 ) \\ &\quad\Leftrightarrow\quad \lambda (\lambda-1 )< 0. \end{aligned}$$

So f decreases on \((1-\lambda,\infty )\) for \(0<\lambda<1\). Noting that \(\beta_{n}=f (n+c )\) (\(c\geq0\)), we have

$$\beta_{1}\geq\beta_{2}\geq\cdots\geq\beta_{n}\geq \delta_{n}, $$

which completes the verification of condition (2.4). Then expansion (3.1) follows from Lemma 2.1.

Since \(a (k,n )\geq0\), by inequality (2.7) we have

$$\bigl\vert \hat{P}_{n}^{\lambda} (x;a,b,c ) \bigr\vert \leq \sum_{k=0}^{n}a (k,n ) \bigl\vert \hat{C}_{n}^{\lambda} (x,c ) \bigr\vert \leq\sum _{k=0}^{n}a (k,n )\hat{C}_{n}^{\lambda} (1,c ) =\hat{P}_{n}^{\lambda} (1;a,b,c ). $$

This completes the proof of the case \(0<\lambda<1\).

Next, we consider the case \(\lambda\geq1\). Note that for \(\lambda=1\) in (2.6), it becomes

$$ x {\hat{C}_{n}^{1} (x,c )} = \hat{C}_{n+1}^{1} (x,c ) +\frac{1}{4} \hat{C}_{n-1}^{1} (x,c ), $$

which still defines a set of (particular) associated ultraspherical polynomials. Let us demonstrate that

$$ \hat{P}_{n}^{\lambda} (x;a,b,c ) =\sum _{k=0}^{n}\bar{a} (k,n )\hat{C}_{n}^{1} (x,c ),\quad \text{where $\bar{a} (k,n )\geq0$.} $$
(3.2)

As mentioned before, condition (2.3) is straightforwardly satisfied. To show that \(\frac{1}{4}=\beta_{k}\geq\delta_{n}\), we only need to require that

$$\frac{1}{4}\geq\frac{ (n+c ) (n+2\lambda+c-1 )}{4 (n+\lambda+c ) (n+\lambda+c-1 )}\quad(\geq\delta_{n}). $$

After little computation, the just mentioned inequality can be simplified to to \(\lambda (\lambda-1 )\geq0\), which suggests the condition \(\lambda\geq1\). This validates expansion (3.2).

Finally, we have

$$\bigl\vert \hat{P}_{n}^{\lambda} (x;a,b,c ) \bigr\vert \leq \sum_{k=0}^{n} {\bar{a}} (k,n ) \bigl\vert \hat {C}_{n}^{1} (x,c ) \bigr\vert \leq\sum_{k=0}^{n} {\bar{a}} (k,n )\hat {C}_{n}^{1} (1,c ) =\hat{P}_{n}^{\lambda} (1;a,b,c ). $$

Proof of Theorem 1.2

To establish inequality (1.5), we first let \(a_{1}=-n\), \(a_{2}=c+\lambda+\mathrm{i}\varPhi\), \(b_{1}=c+1\), and \(b_{2}=c+2\lambda\) in (2.12), and then in view of the defining expression (2.14) of the Laguerre polynomials, we obtain

F22[n,c+λ+iΦc+1,c+2λ;z]=n!(c+1)nΓ(c+2λ)Γ(c+λ+iΦ)Γ(λiΦ)×01tc+λ+iΦ1(1t)λiΦ1Ln(c)(zt)dt

for \(c>0\) and \(\lambda>0\). Substituting this expression of \({}_{2}F_{2}\) into (2.10) and simplifying the resulting equation by using (2.8), we obtain

$$\begin{aligned} &P_{n}^{\lambda} (\cos\theta;a,b,c ) \\ &\quad= \frac{ (c+2\lambda )_{n}}{ (c+1 )_{n}}\mathrm{e}^{\mathrm {i}n\theta} \\ &\qquad\times \int_{ (0,1 )\times\mathbb{R}_{+}^{2}} \biggl[ \int_{0}^{1}L_{n}^{(c)} \bigl(tu_{1}u_{3}+u_{2}u_{3} \bigl(1- \mathrm {e}^{-2\mathrm{i}\theta} \bigr) \bigr)\,\mathrm{d}\mu_{c+\lambda+\mathrm {i}\varPhi,\lambda-\mathrm{i}\varPhi} (u_{3} ) \biggr] \,\mathrm{d}\varPi (t,u_{1},u_{2} ), \end{aligned}$$
(4.1)

where \(\mathrm{d}\varPi (t,u_{1},u_{2} )\) is given by (2.9). This assertion means that the associated Pollaczek polynomials can alternatively be obtained by integrating the Laguerre polynomials.

Next, we choose \(\alpha=c\), \(\lambda=u_{3}\), and \(x=tu_{1}+u_{2} (1-\mathrm {e}^{-2\mathrm{i}\theta} )\) in (2.16), so that the Laguerre polynomials involved in (4.1) can be rewritten as

$$\begin{aligned} L_{n}^{(c)} \bigl(tu_{1}u_{3}+u_{2}u_{3} \bigl(1-\mathrm{e}^{-2\mathrm{i}\theta } \bigr) \bigr) &=\sum _{\ell=0}^{n}\binom{n}{\ell}u_{3}^{\ell}(1-u_{3} )^{n-\ell }\frac{L_{n}^{(c)} (0 )}{L_{\ell}^{(c)} (0 )} \\ &\quad \times L_{\ell}^{(c)} \bigl(tu_{1}+u_{2} \bigl(1-\mathrm{e}^{-2\mathrm{i}\theta} \bigr) \bigr). \end{aligned}$$
(4.2)

Let us assume that \(c=c_{1}+c_{2}+1\) (\(c_{2}>0\), \(c_{1}+1>0\)) for convenience. Then by using (2.15) we get

$$\begin{aligned} L_{\ell}^{(c)} \bigl(tu_{1}+u_{2} \bigl(1-\mathrm{e}^{-2\mathrm{i}\theta} \bigr) \bigr) &=L_{\ell}^{(c_{2}+c_{1}+1)} \bigl(u_{2} \bigl(1-\mathrm{e}^{-2\mathrm{i}\theta } \bigr)+tu_{1} \bigr) \\ &=\sum_{m=0}^{\ell}L_{m}^{(c_{2})} \bigl(u_{2} \bigl(1-\mathrm{e}^{-2\mathrm {i}\theta} \bigr) \bigr)L_{\ell-m}^{(c_{1})} (tu_{1} ). \end{aligned}$$
(4.3)

Further, applying (2.16) to \(L_{m}^{(c_{2})} (u_{2} (1-\mathrm{e}^{-2\mathrm {i}\theta} ) )\), we have

$$ L_{m}^{(c_{2})} \bigl(u_{2} \bigl(1- \mathrm{e}^{-2\mathrm{i}\theta} \bigr) \bigr) =\sum_{k=0}^{m} \binom{m}{k} \bigl(1-\mathrm{e}^{-2\mathrm{i}\theta} \bigr)^{k} \mathrm{e}^{-2\mathrm{i} (m-k )\theta} \frac{L_{m}^{(c_{2})} (0 )}{L_{k}^{(c_{2})} (0 )}L_{k}^{(c_{2})} (u_{2} ), $$
(4.4)

and hence combining equations (4.2), (4.3), and (4.4) suitably, we obtain

$$\begin{aligned} &L_{n}^{(c)} \bigl(tu_{1}u_{3}+u_{2}u_{3} \bigl(1-\mathrm{e}^{-2\mathrm{i}\theta } \bigr) \bigr) \\ &\quad =\sum _{\ell=0}^{n}\binom{n}{\ell}u_{3}^{\ell}(1-u_{3} )^{n-\ell }\frac{L_{n}^{(c)} (0 )}{L_{\ell}^{(c)} (0 )} \\ & \qquad\times\sum_{m=0}^{\ell} L_{\ell-m}^{(c_{1})} (tu_{1} ) \sum _{k=0}^{m}\binom{m}{k} \bigl(1- \mathrm{e}^{-2\mathrm{i}\theta} \bigr)^{k}\mathrm{e}^{-2\mathrm{i} (m-k )\theta} \frac{L_{m}^{(c_{2})} (0 )}{L_{k}^{(c_{2})} (0 )}L_{k}^{(c_{2})} (u_{2} ). \end{aligned}$$
(4.5)

From (4.5) we have

$$\begin{aligned} & \int_{0}^{1}L_{n}^{(c)} \bigl(tu_{1}u_{3}+u_{2}u_{3} \bigl(1- \mathrm{e}^{-2\mathrm {i}\theta} \bigr) \bigr)\,\mathrm{d}\mu_{c+\lambda+\mathrm{i}\varPhi ,\lambda-\mathrm{i}\varPhi} (u_{3} ) \\ &\quad =\sum_{\ell=0}^{n}\binom{n}{\ell} \int_{0}^{1}u_{3}^{\ell}(1-u_{3} )^{n-\ell}\,\mathrm{d}\mu_{c+\lambda +\mathrm{i}\varPhi,\lambda-\mathrm{i}\varPhi} (u_{3} ) \frac{L_{n}^{(c)} (0 )}{L_{\ell}^{(c)} (0 )} \\ &\qquad \times\sum_{m=0}^{\ell} L_{\ell-m}^{(c_{1})} (tu_{1} ) \sum _{k=0}^{m}\binom{m}{k} \bigl(1- \mathrm{e}^{-2\mathrm{i}\theta} \bigr)^{k}\mathrm{e}^{-2\mathrm{i} (m-k )\theta} \frac{L_{m}^{(c_{2})} (0 )}{L_{k}^{(c_{2})} (0 )}L_{k}^{(c_{2})} (u_{2} ) \\ & \quad=\frac{ (c+1 )_{n}}{ (c+2\lambda )_{n} n!}\sum_{\ell =0}^{n} \binom{n}{\ell} \frac{ (c+\lambda+\mathrm{i}\varPhi )_{\ell} (\lambda -\mathrm{i}\varPhi )_{n-\ell}}{L_{\ell}^{(c)} (0 )} \\ &\qquad \times\sum_{m=0}^{\ell} L_{\ell-m}^{(c_{1})} (tu_{1} ) \sum _{k=0}^{m}\binom{m}{k} \bigl(1- \mathrm{e}^{-2\mathrm{i}\theta} \bigr)^{k}\mathrm{e}^{-2\mathrm{i} (m-k )\theta} \frac{L_{m}^{(c_{2})} (0 )}{L_{k}^{(c_{2})} (0 )}L_{k}^{(c_{2})} (u_{2} ). \end{aligned}$$

Thus

$$\begin{aligned} &P_{n}^{\lambda} (\cos\theta;a,b,c ) \\ &\quad=\frac{\mathrm{e}^{\mathrm{i}n\theta}}{n!}\sum _{\ell=0}^{n}\binom {n}{\ell} \frac{ (c+\lambda+\mathrm{i}\varPhi )_{\ell} (\lambda -\mathrm{i}\varPhi )_{n-\ell}}{L_{\ell}^{(c)} (0 )} \\ &\qquad\times \int_{ (0,1 )\times\mathbb{R}_{+}^{2}} \Biggl[\sum_{m=0}^{\ell} L_{\ell-m}^{(c_{1})} (tu_{1} ) \\ &\qquad\times\sum _{k=0}^{m}\binom{m}{k} \bigl(1- \mathrm{e}^{-2\mathrm{i}\theta} \bigr)^{k}\mathrm{e}^{-2\mathrm{i} (m-k )\theta} \frac{L_{m}^{(c_{2})} (0 )}{L_{k}^{(c_{2})} (0 )}L_{k}^{(c_{2})} (u_{2} ) \Biggr] \,\mathrm{d}\varPi (t,u_{1},u_{2} ). \end{aligned}$$

Now we need to carry out some evaluations to obtain a more accurate estimate for the associated Pollaczek polynomials. It is easy to observe from [8, p. 810, Eq. (11)] that

$$ \int_{\mathbb{R}_{+}}L_{k}^{(c_{2})} (u_{2} ) \,\mathrm{d}\mathbf {m}_{1} (u_{2} ) = \int_{\mathbb{R}_{+}}\mathrm{e}^{-u_{2}}L_{k}^{(c_{2})} (u_{2} )\,\mathrm{d}u_{2} =\frac{ (c_{2} )_{k}}{k!}. $$

Then we have

$$\begin{aligned} P_{n}^{\lambda} (\cos\theta;a,b,c ) &=\frac{\mathrm{e}^{\mathrm{i}n\theta}}{n!}\sum _{\ell=0}^{n}\binom {n}{\ell} \frac{ (c+\lambda+\mathrm{i}\varPhi )_{\ell} (\lambda -\mathrm{i}\varPhi )_{n-\ell}}{L_{\ell}^{(c)} (0 )} \\ & \quad\times \sum_{m=0}^{\ell}\frac{ (c_{2}+1 )_{m}}{m!} \int_{ (0,1 )\times\mathbb{R}_{+}} L_{\ell-m}^{(c_{1})} (tu_{1} ) \,\mathrm{d}\mu_{c,\lambda+\mathrm {i}\varPhi} (t )\,\mathrm{d}\mathfrak{m}_{c+2\lambda-1} (u_{1} ) \\ & \quad\times\sum_{k=0}^{m}\binom{m}{k} \bigl(1-\mathrm{e}^{-2\mathrm{i}\theta } \bigr)^{k}\mathrm{e}^{-2\mathrm{i} (m-k )\theta} \frac{ (c_{2} )_{k}}{ (c_{2}+1 )_{k}}. \end{aligned}$$

Therefore it follows that

$$\begin{aligned} & \bigl\vert P_{n}^{\lambda} (\cos\theta;a,b,c ) \bigr\vert \\ &\quad\leq\frac{1}{n!}\sum_{\ell=0}^{n} \binom{n}{\ell} \frac{ \vert (c+\lambda+\mathrm{i}\varPhi )_{\ell} \vert \vert (\lambda-\mathrm{i}\varPhi )_{n-\ell} \vert }{L_{\ell }^{(c)} (0 )} \\ &\qquad \times \sum_{m=0}^{\ell}\frac{ (c_{2}+1 )_{m}}{m!} \biggl\vert \int_{ (0,1 )\times\mathbb{R}_{+}} L_{\ell-m}^{(c_{1})} (tu_{1} ) \,\mathrm{d}\mu_{c,\lambda+\mathrm {i}\varPhi} (t )\,\mathrm{d}\mathfrak{m}_{c+2\lambda-1} (u_{1} ) \biggr\vert \\ & \qquad\times\sum_{k=0}^{m}\binom{m}{k} (2 \sin\theta )^{k} \frac{ (c_{2} )_{k}}{ (c_{2}+1 )_{k}} \\ &\quad\leq \frac{1}{n!}\sum_{\ell=0}^{n} \binom{n}{\ell} \frac{ \vert (c+\lambda+\mathrm{i}\varPhi )_{\ell} \vert \vert (\lambda-\mathrm{i}\varPhi )_{n-\ell} \vert }{L_{\ell }^{(c)} (0 )}\sum_{m=0}^{\ell} \frac{ (c_{2}+1 )_{m}}{m!} (1+2\sin\theta )^{m} \\ &\qquad \times \biggl\vert \int_{ (0,1 )\times\mathbb{R}_{+}} L_{\ell-m}^{(c_{1})} (tu_{1} ) \,\mathrm{d}\mu_{c,\lambda+\mathrm {i}\varPhi} (t )\,\mathrm{d}\mathfrak{m}_{c+2\lambda-1} (u_{1} ) \biggr\vert . \end{aligned}$$

To estimate

$$\biggl\vert \int_{ (0,1 )\times\mathbb{R}_{+}} L_{\ell-m}^{(c_{1})} (tu_{1} ) \,\mathrm{d}\mu_{c,\lambda+\mathrm {i}\varPhi} (t )\,\mathrm{d}\mathfrak{m}_{c+2\lambda-1} (u_{1} ) \biggr\vert , $$

we first define

$$H (t ):= \int_{\mathbb{R}_{+}}L_{\ell-m}^{(c_{1})} (tu_{1} ) \,\mathrm{d}\mathbf{m}_{c+2\lambda-1} (u_{1} ) $$

and observe that

$$\begin{aligned} \bigl\vert H (t ) \bigr\vert &= \biggl\vert \int_{\mathbb{R}_{+}}L_{\ell-m}^{(c_{1})} (tu_{1} ) \,\mathrm{d}\mathbf{m}_{c+2\lambda-1} (u_{1} ) \biggr\vert \\ &\leq \frac{1}{\varGamma (c+2\lambda-1 )} \int_{\mathbb{R}_{+}}u_{1}^{c+2\lambda-2}\mathrm{e}^{-u_{1}} \bigl\vert L_{\ell -m}^{(c_{1})} (tu_{1} ) \bigr\vert \,\mathrm{d}u_{1}. \end{aligned}$$

Using (2.18), we have

$$\begin{aligned} \bigl\vert H (t ) \bigr\vert &\leq \frac{1}{\varGamma (c+2\lambda-1 )} \int_{\mathbb{R}_{+}}u_{1}^{c+2\lambda-2}\mathrm{e}^{-u_{1}} \sum_{k=0}^{\ell-m}\frac{ (c_{1}+1 )_{\ell-m-k}}{ (\ell -m-k )!} \frac{ (tu_{1} )^{k}}{k!} \,\mathrm{d}u_{1} \\ &=\sum_{k=0}^{\ell-m}\frac{ (c_{1}+1 )_{\ell-m-k} (c+2\lambda -1 )_{k}}{ (\ell-m-k )!} \frac{t^{k}}{k!}. \end{aligned}$$

Then

$$\begin{aligned} & \biggl\vert \int_{ (0,1 )\times\mathbb{R}_{+}} L_{\ell-m}^{(c_{1})} (tu_{1} ) \,\mathrm{d}\mu_{c,\lambda+\mathrm {i}\varPhi} (t )\,\mathrm{d}\mathfrak{m}_{c+2\lambda-1} (u_{1} ) \biggr\vert \\ &\quad = \biggl\vert \int_{0}^{1} H (t )\,\mathrm{d}\mu_{c,\lambda+\mathrm{i}\varPhi} (t ) \biggr\vert \\ & \quad\leq \biggl\vert \frac{\varGamma (c+\lambda+\mathrm{i}\varPhi )}{\varGamma (c )\varGamma (\lambda+\mathrm{i}\varPhi )} \biggr\vert \int_{0}^{1} \bigl\vert H (t ) \bigr\vert t^{c-1} (1-t )^{\lambda-1} \,\mathrm{d}t \\ & \quad\leq \biggl\vert \frac{\varGamma (c+\lambda+\mathrm{i}\varPhi )}{\varGamma (c )\varGamma (\lambda+\mathrm{i}\varPhi )} \biggr\vert \sum _{k=0}^{\ell-m}\frac{ (c_{1}+1 )_{\ell-m-k} (c+2\lambda -1 )_{k}}{ (\ell-m-k )! k!} \int_{0}^{1} t^{c+k-1} (1-t )^{\lambda-1} \,\mathrm{d}t \\ & \quad=\frac{\varGamma (\lambda )}{\varGamma (c+\lambda )} \biggl\vert \frac{\varGamma (c+\lambda+\mathrm{i}\varPhi )}{\varGamma (\lambda+\mathrm{i}\varPhi )} \biggr\vert \sum _{k=0}^{\ell-m}\frac{ (c_{1}+1 )_{\ell-m-k} (c+2\lambda -1 )_{k}}{ (\ell-m-k )! k!} \frac{ (c )_{k}}{ (c+\lambda )_{k}} \\ &\quad \leq\frac{\varGamma (\lambda )}{\varGamma (c+\lambda )} \biggl\vert \frac{\varGamma (c+\lambda+\mathrm{i}\varPhi )}{\varGamma (\lambda+\mathrm{i}\varPhi )} \biggr\vert \frac{1}{ (\ell-m )!} \sum_{k=0}^{\ell-m} \binom{\ell-m}{k} (c_{1}+1 )_{\ell-m-k} (c+2\lambda-1 )_{k} \\ & \quad=\frac{\varGamma (\lambda )}{\varGamma (c+\lambda )} \biggl\vert \frac{\varGamma (c+\lambda+\mathrm{i}\varPhi )}{\varGamma (\lambda+\mathrm{i}\varPhi )} \biggr\vert \frac{ (c+2\lambda+c_{1} )_{\ell-m}}{ (\ell-m )!}, \end{aligned}$$

where we have used the Chu–Vandermonde identity (2.13) to get the last equality.

Therefore we have

$$\begin{aligned} & \bigl\vert P_{n}^{\lambda} (\cos\theta;a,b,c ) \bigr\vert \\ &\quad\leq \frac{\varGamma (\lambda )}{\varGamma (c+\lambda )} \biggl\vert \frac{\varGamma (c+\lambda+\mathrm{i}\varPhi )}{\varGamma (\lambda+\mathrm{i}\varPhi )} \biggr\vert \frac {1}{n!}\sum_{\ell=0}^{n} \binom{n}{\ell} \frac{ \vert (c+\lambda+\mathrm{i}\varPhi )_{\ell} \vert \vert (\lambda-\mathrm{i}\varPhi )_{n-\ell} \vert }{L_{\ell }^{(c)} (0 )} \\ & \qquad\times\sum_{m=0}^{\ell}\frac{ (c_{2}+1 )_{m}}{m!} \frac{ (c+2\lambda+c_{1} )_{\ell-m}}{ (\ell-m )!} (1+2\sin \theta )^{m} \\ &\quad\leq \frac{\varGamma (\lambda )\varGamma (\lambda )}{ \vert \varGamma (\lambda+\mathrm{i}\varPhi )\varGamma (\lambda -\mathrm{i}\varPhi ) \vert } \frac{1}{n!}\sum_{\ell=0}^{n} \binom{n}{\ell} \frac{ (c+\lambda )_{\ell} (\lambda )_{n-\ell }}{L_{\ell}^{(c)} (0 )} \\ &\qquad \times\sum_{m=0}^{\ell}\frac{ (c_{2}+1 )_{m}}{m!} \frac{ (c+2\lambda+c_{1} )_{\ell-m}}{ (\ell-m )!} (1+2\sin \theta )^{m} \\ &\quad\leq \frac{\varGamma (\lambda )\varGamma (\lambda )}{ \vert \varGamma (\lambda+\mathrm{i}\varPhi ) \vert ^{2}} \frac{1}{n!}\sum_{\ell=0}^{n} \binom{n}{\ell} \frac{ (c+\lambda )_{\ell} (\lambda )_{n-\ell}}{ (c+1 )_{\ell}} (1+2\sin\theta )^{\ell} \\ &\qquad \times\sum_{m=0}^{\ell}\binom{\ell}{m} (c_{2}+1 )_{m} (c+2\lambda+c_{1} )_{\ell-m} \\ &\quad\leq\frac{\varGamma (\lambda )\varGamma (\lambda )}{ \vert \varGamma (\lambda+\mathrm{i}\varPhi ) \vert ^{2}} \frac{1}{n!} (1+2\sin\theta )^{n}\sum _{\ell=0}^{n}\binom {n}{\ell} \frac{ (c+\lambda )_{\ell} (\lambda )_{n-\ell}}{ (c+1 )_{\ell}} (2c+2\lambda )_{\ell} \\ &\quad\leq\frac{\varGamma (\lambda )\varGamma (\lambda )}{ \vert \varGamma (\lambda+\mathrm{i}\varPhi ) \vert ^{2}} \frac{ (2c+2\lambda )_{n}}{ (c+1 )_{n} n!} (1+2\sin\theta )^{n} \sum _{\ell=0}^{n}\binom{n}{\ell} (c+\lambda )_{\ell} (\lambda )_{n-\ell}. \end{aligned}$$
(4.6)

To verify the last inequality in (4.6), we prove that the sequence \(\{ (2c+2\lambda )_{\ell}/ (c+1 )_{\ell}\}\) is increasing for \(\ell=0,1,\ldots, n\). Consider the function

$$ f (x ):=\log\frac{\varGamma (2c+2\lambda+x )}{\varGamma (c+1+x )},\quad x\in[0,\infty). $$

Taking the derivative with respect to x, we obtain

$$f' (x ):=\psi (2c+2\lambda+x )-\psi (c+1+x ), $$

where \(\psi (z )\) denotes the psi function (or digamma function). Since \(c+2\lambda>1\), from the monotonicity of the psi function we have that \(f' (x )>0\). Thus

$$\frac{\varGamma (2c+2\lambda+x )}{\varGamma (c+1+x )}, $$

as a function of x, is increasing on \([0,\infty)\). So the sequence

$$\frac{ (2c+2\lambda )_{\ell}}{ (c+1 )_{\ell}} =\frac{\varGamma (c+1 )}{\varGamma (2c+2\lambda )}\frac{\varGamma (2c+2\lambda+\ell )}{\varGamma (c+1+\ell )} $$

also increases on \(\{0,1,\ldots,n\}\).

The result (1.5) finally follows by the Chu–Vandermonde identity (2.13).

Remarks and observations

By letting \(c=0\) in Theorem 1.1 we obtain the following result:

Corollary 5.1

([20, p. 4])

For \(a,b\geq0\), \(\lambda>0\), and \(x\in[-1,1]\), we have

$$ \bigl\vert P_{n}^{\lambda} (x;a,b ) \bigr\vert \leq P_{n}^{\lambda} (1;a,b ). $$
(5.1)

Note that Yadav’s inequality (5.1) generalizes Askey’s inequalities obtained in [2] and [3].

  1. (i)

    Note that inequality (5.1) can be equivalently written as

    $$\bigl\vert P_{n}^{\lambda}(x;a,b ) \bigr\vert \leq L_{n}^{ (2\lambda -1 )} \bigl(-2 (a+b ) \bigr), $$

    since \(P_{n}^{\lambda}(1;a,b )=L_{n}^{ (2\lambda-1 )} (-2 (a+b ) )\), where \(L_{n}^{(\alpha)} (x )\) denotes the Laguerre polynomials defined by (2.14). For the associated Pollaczek polynomials, we have ([19, p. 305])

    $$ P_{n}^{\lambda}(1;a,b,c )=L_{n}^{(2\lambda-1)} \bigl(-2 (a+b );c \bigr), $$

    where \(L_{n}^{(\alpha)} (x;c )\) are the associated Laguerre polynomials (see, e.g., [4] and [19]). Therefore inequality (1.4) can be also expressed as

    $$ \bigl\vert P_{n}^{\lambda}(x;a,b,c ) \bigr\vert \leq L_{n}^{ (2\lambda-1 )} \bigl(-2 (a+b );c \bigr). $$
    (5.2)

    Although these inequalities are quite elegant, they are actually somewhat difficult to use directly, especially for (5.2). The asymptotic behavior of \(L_{n}^{ (\alpha )} (x;c )\) for fixed \(x>0\), although not applicable to our case, can be found in [4, p. 24, Eq. (2.16)].

  2. (ii)

    Note that the associated ultraspherical (Gegenbauer) polynomials can be represented in terms of the associated Pollaczek polynomials as \(C_{n}^{\lambda} (\cos\theta;c )=P_{n}^{\lambda } (\cos\theta;0,0,c )\). Hence Theorem 1.1 gives

    $$ \bigl\vert C_{n}^{\lambda} (\cos\theta;c ) \bigr\vert \leq \frac{ (c+2\lambda )_{n} (2c+2\lambda )_{n}}{ (c+1 )_{n} n!} (1+2\sin\theta )^{n}, $$

    which seems to be a new inequality for the associated ultraspherical polynomials.

  3. (iii)

    Finally, we may mention that a more accurate (and much involved) upper bound than (2.18) for the Laguerre polynomials \(L_{n}^{(\alpha)} (x )\) with \(n\geq2\) can be found in [15, p. 491, Theorem 1]. It is possible to obtain an improvement of Theorem 1.1, which may be not easy by using this inequality [15, p. 491, Theorem 1] because derivations would be quite complicated. However, we do not pursue it here but leave it as a worthwhile problem for the interested reader.

References

  1. 1.

    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1964)

  2. 2.

    Askey, R.: Orthogonal polynomials and positivity. In: Ludwig, D., Olver, F.W.J. (eds.) Special Functions and Wave Propagation. Studies in Applied Mathematics, vol. 6, pp. 64–85. Society for Industrial and Applied Mathematics, Philadelphia (1970)

  3. 3.

    Askey, R.: Orthogonal expansions with positive coefficients II. SIAM J. Math. Anal. 2, 340–346 (1971)

  4. 4.

    Askey, R., Wimp, J.: Associated Laguerre and Hermite polynomials. Proc. R. Soc. Edinb. 96A, 15–37 (1984)

  5. 5.

    Carlson, B.C.: Special Functions of Applied Mathematics. Academic Press, New York (1977)

  6. 6.

    Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon & Breach, New York (1978)

  7. 7.

    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Higher Transcendental Functions, vol. II. McGraw-Hill, New York (1981)

  8. 8.

    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Jeffrey, A., Zwillinger, D. (eds.). Academic Press, New York (2007)

  9. 9.

    Huang, X.-M., Wong, R.: Uniform asymptotics and zeros of the associated Pollaczek polynomials. Stud. Appl. Math. (2020) 1–22. https://doi.org/10.1111/sapm.12292

  10. 10.

    Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge (2009)

  11. 11.

    Lewandowski, Z., Szynal, J.: An upper bound for the Laguerre polynomials. J. Comput. Appl. Math. 99(1–2), 529–533 (1998)

  12. 12.

    Love, E.R.: Inequalities for Laguerre functions. J. Inequal. Appl. 1(3), 293–299 (1997)

  13. 13.

    Luo, M.-J., Raina, R.K.: Generating functions of Pollaczek polynomials: a revisit. Integral Transforms Spec. Funct. 30, 893–919 (2019)

  14. 14.

    Luo, M.-J., Wong, R.: Asymptotics of the associated Pollaczek polynomials. Proc. Am. Math. Soc. 147(6), 2583–2597 (2019)

  15. 15.

    Michalska, M., Szynal, J.: A new bound for the Laguerre polynomials. J. Comput. Appl. Math. 133(1–2), 489–493 (2001)

  16. 16.

    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)

  17. 17.

    Pollaczek, F.: Sur une famille de polynômes orthogonaux à quatre paramètres. C. R. Acad. Sci. Paris 230, 2254–2256 (1950)

  18. 18.

    Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted Press, Chichester; Wiley, New York (1984)

  19. 19.

    Wimp, J.: Pollaczek polynomials and Padé approximants: some closed-form expressions. J. Comput. Appl. Math. 32(1–2), 301–310 (1990)

  20. 20.

    Yadav, S.P.: Some inequalities for Pollaczek polynomials. Jñānābha 11, 1–4 (1981)

Download references

Acknowledgements

Authors are thankful to the referees for their useful suggestions.

Availability of data and materials

Not applicable.

Funding

The research of the first author is sponsored by Shanghai Sailing Program (No. 19YF1400100), Science and Technology Commission of Shanghai Municipality (No. 18dz2271000) and the Initial Research Funds for Young Teachers of Donghua University (No. 109-07-0053038).

Author information

All authors contributed equally to the manuscript. Both authors read and approved the final manuscript.

Correspondence to Min-Jie Luo.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Luo, M., Raina, R.K. Two inequalities for the associated Pollaczek polynomials. J Inequal Appl 2020, 9 (2020). https://doi.org/10.1186/s13660-020-2282-5

Download citation

MSC

  • 33C45
  • 26D15
  • 33C20

Keywords

  • Inequality
  • Laguerre polynomial
  • Pollaczek polynomial