Abstract
Gegenbauer, also known as ultra-spherical, polynomials appear often in numerical analysis or interpolation. In the present text we find a recursive formula for and compute the asymptotic behavior of their \(L^2\)-norm.
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Introduction
Gegenbauer polynomials \(\mathcal {C}_{n}^{(\lambda )}\), where \(\lambda \in I_G:=\,(-\frac{1}{2},0)\cup (0,\infty )\) is called the index and \(n\in \mathbb {N}_0\) is the degree, are the coefficients of following power series expansion in \(\alpha\):
The case \(\lambda =0\) is not considered here. \(\{\mathcal {C}_{n}^{(\lambda )}\}_{n\in \mathbb {N}_0}\) are orthogonal with respect to the measure \((1-x^2)^{\lambda -1/2}\ \mathrm {d}x\) over \([-1,1]\), and by [4] [Eq. 8.930]:
In the tables [4] by Gradshteyn and Ryzhik five pages are devoted to various integrals of Gegenbauer polynomials, for they appear in many branches of mathematics and theoretical physics, but not all cases of interest are covered: in applications to meson physics in [5], integrals of the type
had to be evaluated for \(\ell \in \mathbb {N}_0\), \(p\in \mathbb {Z}\) and \(\mathrm {Re}(\alpha -\frac{1}{2}-p)>0\), where the notation follows [6]. Later, Rashid obtained in [6] a general identity for
in terms of a hyper-geometric series\(\ _4F_3\) evaluated at 1 for \(\ell ,k\in \mathbb {N}_0\), \(p\in \mathbb {Z}\) and \(\mathrm {Re}\big ((\alpha +\beta -1)/2-p\big )>0\). Due to the close connection of zonal harmonics to Gegenbauer polynomials, similar integrals also appear in applications of determinantal point processes to energy estimates on spheres [2]. Yet the basic question of the \(L^2\)-norm of these polynomials has not been addressed in the literature, and we will fill this gap. The following notation will be used:
We derive an asymptotic formula for \(\Vert \mathcal {C}_{n}^{(\lambda )}\Vert ^2_2\) when \(\lambda >0\) in Proposition 1. Indeed, one of the key ingredients in [1] was the asymptotic nature of \(\Vert \mathcal {C}_{n}^{(2)}\Vert ^2_2\) in n, and the following lemma was proved in [1][Lemmas 6.1 and 6.2]:
Lemma 1
Let \(\psi\) denote the digamma function and \(\gamma\) the Euler–Mascheroni constant. Then the Gegenbauer polynomials satisfy for \(n\ge 2\):
The following result of Corollary 5.2 from [3] will prove to be indispensable.
Theorem 1
(Dette [3]) The Gegenbauer polynomials satisfy for \(\lambda \in I_G\)
Our main theorem is as follows, and we will use it to derive the asymptotic behavior of \(\Vert \mathcal {C}_{n}^{(\lambda )}\Vert ^2_2\).
Theorem 2
(Main Result) The Gegenbauer polynomials satisfy for \(\lambda \in I_G\):
From Theorem 2 we can deduce following more compact formulas.
Corollary 1
The Gegenbauer polynomials satisfy for \(\lambda \in I_G\) and \(n>0\):
Corollary 2
The Gegenbauer polynomials satisfy for \(\lambda \in I_G\) and \(n>1\):
Proposition 1
Let \(\mathcal {B}(x,y)\) denote the beta function. The following asymptotic formulas in n hold for \(\lambda \in (0,1)\) and \(\delta =\max \{4\lambda -1,2\lambda \}\):
The following asymptotic formulas hold for \(\lambda >1\) and \(\rho =\max \{4\lambda -3,2\lambda \}\):
The identity \(2\cdot \Vert \mathcal {C}_{n}^{(1)}\Vert _2^2=\psi (n+\tfrac{3}{2})+\gamma +\log (4)\) is given in [1][Eq. 14].
Ingredients for the Proof of the Theorem
In this section we collect known results concerning Gegenbauer polynomials for later reference and the reader’s convenience, and we derive some technical lemmas in Subsect. 2.1 to prove Theorem 2. To avoid repetition, we will assume \(\lambda \in I_G\) for the rest of the text if not stated otherwise. Note first that
and \(\mathcal {C}_{n}^{(\lambda )}(1)\) is the maximum on \([-1,1]\) for \(\lambda >0\) by [7][Eq. 7.33.1]. Also, by (4):
Identities for Gegenbauer polynomials
Lemma 2
The Gegenbauer polynomials satisfy following identities:
Proof
First we use (6) and apply (5) to the right-hand side below proving \((\star )\) while using the short-hand \(\ell :=\lambda +1\):
Next we obtain by the binomial theorem with (6), \((\star )\) and (4)
Integration by parts then finishes the argument. \(\square\)
Lemma 3
The Gegenbauer polynomials satisfy the following identity:
Proof
Let \(n=2m\). By Lemma 2 and a telescoping sum argument:
Using (1) and summing up, and an application of Dette’s result (2) yields:
The case \(n+1=2m\) is analogous.\(\square\)
Lemma 4
The Gegenbauer polynomials satisfy the following identity:
Proof
Note first that by (5) and by quadratic completion
Hence by the binomial theorem and again by (5)
which proves the result when we substitute (7) and use (5) one last time.\(\square\)
Proof of the main results
Proof of Theorem 2
Subtract the left-hand sides of Lemma 3 and Lemma 4:
an application of Dette’s formula (2) then gives the desired expression.\(\square\)
Proof of Corollary 1
We use Lemma 4, add zero and obtain with Theorem 2
We re-order to obtain the result.\(\square\)
Proof of Corollary 2
This follows directly from the proofs of Theorem 2 and Corollary 1.\(\square\)
Remark 1
For our asymptotic analysis we will need the following identity, see [8]: For \(|z|\rightarrow \infty\) and \(\alpha ,\beta \ge 0\):
we obtain by (4) for \(\lambda >0\):
Proof of Proposition 1
We will write \(\Vert \mathcal {C}_{n}^{(\lambda )}\Vert _2^2=\Theta (n^{\Phi (\lambda )})\) if there are some constants \(c_1,c_2>0\) such that \(c_1n^{\Phi (\lambda )}\le \Vert \mathcal {C}_{n}^{(\lambda )}\Vert _2^2 \le c_2n^{\Phi (\lambda )}\) for all n big enough. First we use (3) to show by induction that \(\Phi (\lambda )\) exists for \(\lambda >1\), and that \(\big [\mathcal {C}_{n}^{(\lambda )}(1)\big ]^2=\Theta (n^{\Phi (\lambda )+2})\).
The case \(\lambda =m\in \mathbb {N}_{>1}\): Lemma 1 gives the result for \(\lambda =2\), and if it holds for m, then with (3) and abuse of notation we have:
This proves the claim as it shows that \(\Vert \mathcal {C}_{n}^{(m+1)}\Vert _2^2=\Theta (n^{\Phi (m)+4})\), but by (4):
which, when squared and \(\lambda =m\), is of order \(\Phi (m)+6.\)
The case \(\lambda \in (m,m+1)\) for \(m\in \mathbb {N}\): For \(\lambda \in (0,1)\) and \(\theta \in [0,\pi ]\):
We square this inequality, multiply by \(\sin (\theta )^{1-2\lambda }\) and integrate:
where we used a change of variables \(\theta =\arcsin (x)\) and \(\mathcal {B}(x,y)\) is the beta function. This in combination with (10) and (3) gives for \(\delta =\max \{4\lambda -1,2\lambda \}\):
Thus for \(\lambda \in (0,1)\): \(\Phi (\lambda +1)=4\lambda\), and \([\mathcal {C}_{n}^{(\lambda +1)}(1)]^2=\Theta \big (n^{4\lambda +2}\big )\) by (10), which finishes the case for the interval (1, 2), and we use induction with (3) and (11).
Hence, the two leading terms in the asymptotic form of \(\Vert \mathcal {C}_{n}^{(\lambda +1)}\Vert _2^2\) are in the expansion of \(\mathcal {C}_{n}^{(\lambda )}(1)\) when \(\lambda >1\); using once more (10) and (3) yields
The asymptotic of the rest term follows by (3), Equation (12) and induction for non-integer \(\lambda \in \mathbb {R}_{>2}\) or else Lemma 1 and induction when \(\lambda \in \mathbb {N}_{> 2}\). These asymptotic formulas in combination with Corollary 1 and (10) will now finish the argument. We will only do the case \(\lambda >1\), for the case \(\lambda \in (0,1)\) is similar; let \(\rho =\max \{4\lambda -3,2\lambda \}\):
\(\square\)
Remark 2
One can use Lemma 1, Corollary 1 and Corollary 2 to find exact formulas for \(\Vert \sqrt{1-x^2}\ \mathcal {C}_{n}^{(m)}\Vert _2^2\) and \(\Vert \mathcal {C}_{n}^{(m)}\Vert _2^2\) where \(m\in \mathbb {N}_{>1}\).
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Acknowledgements
The help of J. Brauchart, P. Grabner and J. Thuswaldner is gratefully appreciated, who proof read the manuscript, made useful remarks on presentation, made me aware of reference [6] and suggested to generalize Proposition 1 from \(\lambda \in \frac{1}{2}\mathbb {N}\) to \(\lambda >0\). This paper is published open access thanks to the TU Graz Open Access Publishing Fund. I thank the anonymous reviewer for helpful remarks.
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The author thankfully acknowledges support by the Austrian Science Fund (FWF): F5503 “Quasi-Monte Carlo Methods” and FWF: W1230 “Doctoral School Discrete Mathematics,” and the Austrian Marshall Plan Foundation.
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Ferizović, D. On the \(L^2\)-norm of Gegenbauer polynomials. Math Sci 16, 115–119 (2022). https://doi.org/10.1007/s40096-021-00398-1
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DOI: https://doi.org/10.1007/s40096-021-00398-1