Functions on the Plane as Combinations of Powers of Distances to Points

Abstract

Some real functions on the plane can be expressed as a linear combination of powers of distances to certain points.

1. Introduction

In this paper we construct a function basis to express different powers of the distance between a point and the origin. Namely, for two given natural numbers 𝑚 and 𝑛 we find points in the plane such that the function 𝑟^𝑛+2𝑚cos(𝑛𝜃) can be expressed as linear combination of powers of distances to those points (see formula (3.15) below), both the linear combination and the given function having the same domain in the plane (for the notion of special function that we use, see Section 3.1).

One of the important observations in the theory of real functions is that series expansions lead to certain interesting numbers and functions (e.g., Fourier coefficients, 𝐿-functions). Namely, usually series representations can be constructed explicitly in some model spaces of sections of various function bundles over appropriate analytic manifolds. (By considering functions on the plane as maps 𝐹: ℝ²→ℝ, the graphs of these functions are the sections described by the map 𝜓: ℝ²→ℝ³; (𝑥,𝑦) ↦ (𝑥,𝑦, 𝐹(𝑥,𝑦)), which defines 2-sub-manifolds on ℝ³.) (for analytic functions on the plane ℝ²; see Section 2.2).

We denote by 𝒜(ℝ²) the subspace of real analytic functions. It is well known that any polynomial of degree 𝑛 can be written exactly as a linear combination of powers of the right number of terms of the form 𝑥⋅𝑣_𝑘+𝑏_𝑘. We realize a series representation in the space of analytic functions on the plane 𝒜(ℝ²).

Consider a point 𝐩 in the plane ℝ² with coordinates (𝑟cos𝜃,𝑟sin𝜃).

The distance used here is the Euclidean distance. Denote |𝐩|=𝑟. (1.1)

We prove the following theorem.

Theorem 1.1. Consider two natural numbers 𝑚 and 𝑛. There are 𝑁 numbers 𝑎₁,…,𝑎_𝑁 and 𝑁 points 𝐩₁,…,𝐩_𝑁, such that 𝑟^𝑛+2𝑚cos(𝑛𝜃)=^𝑁_𝑖=1𝑎_𝑖𝐩−𝐩_𝑖^2(𝑚+𝑛). (1.2)

Remark 1.2. The numbers 𝑎₁,…,𝑎_𝑁 and the points 𝐩₁,…,𝐩_𝑁 can be determined exactly.

Corollary 1.3. The linear combinations of 𝑟^𝑛+2𝑚cos(𝑛𝜃), 𝑚, 𝑛=0,1,…, can be expressed similarly.

Remark 1.4. An analogous theorem holds for the sine function.

Denote by 𝑥 the maximum integer less than or equal to 𝑥.

Consider the function𝛿𝑥,𝑦=0if𝑥≠𝑦,1if𝑥=𝑦. (1.3)

2. Proof of Theorem 1.1

We will denote by 𝑝_𝑑 the Legendre polynomial of degree 𝑑.

The following formula is formula (8.921) of [1]:1√1−2𝑡𝑥+𝑡²=^∞_𝑑=0𝑡^𝑑𝑝_𝑑(𝑥),|𝑡|<min𝑥±√𝑥²−1. (2.1) Consider the set |𝑡|<min|𝑥±√𝑥²−1| of ℝ². Consider a subset ℋ (not necessarily proper) of such set. We will work solely with analytic functions on ℋ.

Consider the series ∑^∞_𝑖=0∑^𝑖_𝑗=0𝑎_𝑖𝑗𝑡^𝑖+𝑗𝑥^𝑖−𝑗.

Divide this number by √1−2𝑡𝑥+𝑡². By formula (2.1), the following formula holds:∑^∞_𝑖=0∑^𝑖_𝑗=0𝑎_𝑖𝑗𝑡^𝑖+𝑗𝑥^𝑖−𝑗√1−2𝑡𝑥+𝑡²=^∞𝑖_𝑖=0𝑗=0𝑎_𝑖𝑗𝑡^𝑖+𝑗𝑥^𝑖−𝑗⋅^∞_𝑑=0𝑡^𝑑𝑝_𝑑(𝑥). (2.2) We use the formula of the product of two series ∑^∞_𝑛=0𝑎_𝑛 and ∑^∞_𝑛=0𝑏_𝑛:^∞_𝑛=0𝑎_𝑛^∞_𝑛=0𝑏_𝑛=^∞𝑛_𝑛=0𝑘=0𝑎_𝑘𝑏_𝑛−𝑘. (2.3) The right side of formula (2.2) becomes^∞𝑖_𝑖=0𝑗=0𝑎_𝑖𝑗𝑡^𝑖+𝑗𝑥^𝑖−𝑗⋅^∞_𝑑=0𝑡^𝑑𝑝_𝑑(𝑥)=^∞𝑖_𝑖=0𝑘=0^𝑖−𝑘_𝑗=0𝑎_{𝑖−𝑘,𝑗}𝑝_𝑘(𝑥)𝑡^𝑖+𝑗𝑥^{𝑖−𝑘−𝑗}. (2.4) Let us change the order of addition with respect to the indices 𝑗 and 𝑘:^𝑖_𝑘=0^𝑖−𝑘_𝑗=0𝑎_{𝑖−𝑘,𝑗}𝑝_𝑘(𝑥)𝑡^𝑖+𝑗𝑥^{𝑖−𝑘−𝑗}=^𝑖_𝑗=0^𝑖−𝑗_𝑘=0𝑎_{𝑖−𝑘,𝑗}𝑝_𝑘(𝑥)𝑡^𝑖+𝑗𝑥^{𝑖−𝑘−𝑗}. (2.5) The following formula is formula (8.911) of [1]:𝑝_𝑘(𝑥)=^𝑘/2_𝑙=0𝑏_𝑘𝑙𝑥^𝑘−2𝑙, (2.6) where𝑏_𝑘𝑙=12^𝑘(−1)^𝑙(2𝑘−2𝑙)!()!()!.𝑙!𝑘−𝑙𝑘−2𝑙 (2.7) Substituting (2.6) in (2.5),^𝑖_𝑗=0^𝑖−𝑗_𝑘=0𝑎_{𝑖−𝑘,𝑗}𝑝_𝑘(𝑥)𝑡^𝑖+𝑗𝑥^{𝑖−𝑘−𝑗}=^𝑖_𝑗=0^𝑖−𝑗_𝑘=0𝑎_{𝑖−𝑘,𝑗}^𝑘/2_𝑙=0𝑏_𝑘𝑙𝑥^𝑘−2𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑘−𝑗}=^𝑖_𝑗=0^𝑖−𝑗_𝑘=0^𝑘/2_𝑙=0𝑎_{𝑖−𝑘,𝑗}𝑏_𝑘𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}. (2.8) Let us change the order of addition with respect to the indices 𝑘 and 𝑙:^𝑖−𝑗_𝑘=0^𝑘/2_𝑙=0𝑎_{𝑖−𝑘,𝑗}𝑏_𝑘𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}=^{(𝑖−𝑗)/2}_𝑙=0^𝑖−𝑗_𝑘=2𝑙𝑎_{𝑖−𝑘,𝑗}𝑏_𝑘𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}. (2.9) Denote by𝑎_𝑖𝑗𝑙=^𝑖−𝑗_𝑘=2𝑙𝑎_{𝑖−𝑘,𝑗}𝑏_𝑘𝑙. (2.10) The sum (2.9) is ∑^{(𝑖−𝑗)/2}_𝑙=0𝑎_𝑖𝑗𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}. We will substitute this sum in (2.8).^𝑖_𝑗=0^𝑖−𝑗_𝑘=0^𝑘/2_𝑙=0𝑎_{𝑖−𝑘,𝑗}𝑏_𝑘𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}=^𝑖_𝑗=0^{(𝑖−𝑗)/2}_𝑙=0𝑎_𝑖𝑗𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}. (2.11)

Let us change the order of addition with respect to the indices 𝑗 and 𝑙:^𝑖_𝑗=0^{(𝑖−𝑗)/2}_𝑙=0𝑎_𝑖𝑗𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}=^𝑖/2_𝑙=0^𝑖−2𝑙_𝑗=0𝑎_𝑖𝑗𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}. (2.12) Let us add with respect to the index 𝑖 from zero to infinity^∞_𝑖=0^𝑖/2_𝑙=0^𝑖−2𝑙_𝑗=0𝑎_𝑖𝑗𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}. (2.13) Let us change the order of addition with respect to the indices 𝑖 and 𝑙:^∞_𝑖=0^𝑖/2_𝑙=0^𝑖−2𝑙_𝑗=0𝑎_𝑖𝑗𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}=^∞∞_{𝑙=0𝑖=2𝑙}^𝑖−2𝑙_𝑗=0𝑎_𝑖𝑗𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}. (2.14) Change the index 𝑖 for index 𝑖^′=𝑖−2𝑙:^∞∞_{𝑙=0𝑖=2𝑙}^𝑖−2𝑙_𝑗=0𝑎_𝑖𝑗𝑙𝑡^𝑖+𝑗𝑥^{𝑖−𝑗−2𝑙}=^∞∞_𝑙=0^𝑖_𝑖′=0^′_𝑗′=0𝑎_{𝑖′+2𝑙,𝑗′𝑙}𝑡^{𝑖′+2𝑙+𝑗′}𝑥^{𝑖′−𝑗′}. (2.15) We change the indices 𝑖′ and 𝑗′ for the indices 𝑗𝑖′=𝑖−𝑙,_′=𝑗−𝑙,_{∞∞𝑙=0𝑖′𝑖=0′𝑗′=0}𝑎_{𝑖′+2𝑙,𝑗′𝑙}𝑡^{𝑖′+2𝑙+𝑗′}𝑥^{𝑖′−𝑗′}=^∞∞_{𝑙=0𝑖=𝑙}^𝑖−𝑙_𝑗=𝑙𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}𝑡^𝑖+𝑗𝑥^𝑖−𝑗. (2.16) Let us change the order of addition with respect to the indices 𝑖 and 𝑙:^∞∞_{𝑙=0𝑖=𝑙}^𝑖−𝑙_𝑗=𝑙𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}𝑡^𝑖+𝑗𝑥^𝑖−𝑗=^∞𝑖_𝑖=0𝑙=0^𝑖−𝑙_𝑗=𝑙𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}𝑡^𝑖+𝑗𝑥^𝑖−𝑗. (2.17) Let us change the order of addition with respect to the indices 𝑗 and 𝑙:^𝑖_𝑙=0^𝑖−𝑙_𝑗=𝑙𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}𝑡^𝑖+𝑗𝑥^𝑖−𝑗=^𝑖𝑗_𝑗=0𝑙=0𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}𝑡^𝑖+𝑗𝑥^𝑖−𝑗. (2.18) Denote by𝐴_𝑖𝑗=^𝑗_𝑙=0𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}. (2.19) The sum (2.18) becomes^𝑖_𝑙=0^𝑖−𝑙_𝑗=𝑙𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}𝑡^𝑖+𝑗𝑥^𝑖−𝑗=^𝑖_𝑗=0𝐴_𝑖𝑗𝑡^𝑖+𝑗𝑥^𝑖−𝑗. (2.20) Let us add with respect to the index 𝑖 from zero to infinity.

The ratio (2.2) becomes∑^∞_𝑖=0∑^𝑖_𝑗=0𝑎_𝑖𝑗𝑡^𝑖+𝑗𝑥^𝑖−𝑗√1−2𝑡𝑥+𝑡²=^∞𝑖_𝑖=0𝑗=0𝐴_𝑖𝑗𝑡^𝑖+𝑗𝑥^𝑖−𝑗. (2.21)

2.1. Coefficients with Three Indices

Let us calculate the numbers 𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}. By formula (2.10), the following formula holds:𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}=^{𝑖+2𝑙−𝑗}_𝑘=2𝑙𝑎_{𝑖+𝑙−𝑘,𝑗−𝑙}𝑏_𝑘𝑙. (2.22) Change the index 𝑘 for index 𝑘−2𝑙:^{𝑖+2𝑙−𝑗}_𝑘=2𝑙𝑎_{𝑖+𝑙−𝑘,𝑗−𝑙}𝑏_𝑘𝑙=^𝑖−𝑗_𝑘=0𝑎_{𝑖−𝑙−𝑘,𝑗−𝑙}𝑏_{2𝑙+𝑘,𝑙}. (2.23) By formula (2.7), the following formula holds:𝑏_{2𝑙+𝑘,𝑙}=(−1)^𝑙(2𝑙+2𝑘)!2^2𝑙+𝑘().𝑙!𝑙+𝑘!𝑘! (2.24) Let us use the formula𝑑^𝑛𝑑𝑥^𝑛1√𝑥=(−1)^𝑛(2𝑛)!4^−𝑛𝑥^{−𝑛−1/2}𝑛! (2.25) to write(−1)^𝑙(2𝑙+2𝑘)!2^2𝑙+𝑘()=()𝑙!𝑙+𝑘!𝑘!−1^𝑘2^𝑘𝑥^{𝑙+𝑘+1/2}𝑑𝑙!𝑘!𝑑𝑥^{𝑙+𝑘𝑙+𝑘}1√𝑥. (2.26) We will substitute this in (2.23):^𝑖−𝑗_𝑘=0𝑎_{𝑖−𝑙−𝑘,𝑗−𝑙}𝑏_{2𝑙+𝑘,𝑙}=^𝑖−𝑗_𝑘=0𝑎_{𝑖−𝑙−𝑘,𝑗−𝑙}(−1)^𝑘2^𝑘𝑥^{𝑙+𝑘+1/2}𝑑𝑙!𝑘!𝑑𝑥^{𝑙+𝑘𝑙+𝑘}1√𝑥. (2.27) Formula (2.22) becomes𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}=^𝑖−𝑗_𝑘=0𝑎_{𝑖−𝑙−𝑘,𝑗−𝑙}(−1)^𝑘2^𝑘𝑥^{𝑙+𝑘+1/2}𝑑𝑙!𝑘!𝑑𝑥^{𝑙+𝑘𝑙+𝑘}1√𝑥. (2.28)

2.2. A Series of Half-Integer Powers

Consider three numbers 𝑎, 𝑏, and 𝑐.

We will develop the power of the sum of three numbers.

Lemma 2.1. Consider a natural number 𝑛. Then, the following formula holds: (𝑎+𝑏+𝑐)^𝑛=^𝑛𝑖_𝑖=0𝑗=0𝑛!𝑎^𝑛−𝑖𝑏^𝑖−𝑗𝑐^𝑗()!.𝑛−𝑖!𝑗!𝑖−𝑗 (2.29)

Consider the series ∑^∞_𝑛=0𝑎_𝑛(𝑎²−2𝑎𝑏𝑐+𝑐²)^𝑛.

By Lemma 2.1 the following formula holds:^∞_𝑛=0𝑎_𝑛𝑎²−2𝑎𝑏𝑐+𝑐^2𝑛=^∞_𝑛=0𝑎_𝑛^𝑛𝑖_𝑖=0𝑗=0(−1)^𝑖−𝑗2^𝑖−𝑗𝑛!𝑎^{2𝑛−𝑖−𝑗}𝑏^𝑖−𝑗𝑐^𝑖+𝑗()!.𝑛−𝑖!𝑗!𝑖−𝑗 (2.30) We will distribute this sum, and we will exchange the order of addition:^∞_𝑛=0𝑎_𝑛𝑎²−2𝑎𝑏𝑐+𝑐^2𝑛=^∞𝑖_𝑖=0^∞_{𝑗=0𝑛=𝑖}𝑎_𝑛(−1)^𝑖−𝑗2^𝑖−𝑗𝑛!𝑎^{2𝑛−𝑖−𝑗}𝑏^𝑖−𝑗𝑐^𝑖+𝑗()!.𝑛−𝑖!𝑗!𝑖−𝑗 (2.31) Denote by𝑎_𝑖𝑗=^∞_𝑛=𝑖𝑎_𝑛(−1)^𝑖−𝑗2^𝑖−𝑗𝑛!𝑎^{2𝑛−𝑖−𝑗}()!.𝑛−𝑖!𝑗!𝑖−𝑗 (2.32) The series becomes^∞_𝑛=0𝑎_𝑛𝑎²−2𝑎𝑏𝑐+𝑐^2𝑛=^∞𝑖_𝑖=0𝑗=0𝑎_𝑖𝑗𝑏^𝑖−𝑗𝑐^𝑖+𝑗. (2.33) Consider the formula𝑑^𝑖𝑥^𝑛𝑑𝑥^𝑖_𝑥=𝑎2=𝑎^{2𝑛−2𝑖}𝑛!()!.𝑛−𝑖 (2.34)

2.2.1. Formulas for Coefficients with Two Indices

Let us use formula (2.34) to rewrite formula (2.32).𝑎_𝑖𝑗=^∞_𝑛=𝑖𝑎_𝑛(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^𝑖−𝑗!𝑑𝑗!𝑖−𝑗^𝑖𝑥^𝑛𝑑𝑥^𝑖_𝑥=𝑎2. (2.35) Consider the function𝑓(𝑥)=^∞_𝑛=0𝑎_𝑛𝑥^𝑛. (2.36) Let us differentiate the series term by term:𝑑^𝑖𝑑𝑥^𝑖𝑓(𝑥)=^∞_𝑛=0𝑎_𝑛𝑑^𝑖𝑥^𝑛𝑑𝑥^𝑖. (2.37) The 𝑖−1 first terms in this series are equal to zero. Then, the following formula holds:𝑑^𝑖𝑑𝑥^𝑖𝑓(𝑥)=^∞_𝑛=𝑖𝑎_𝑛𝑑^𝑖𝑥^𝑛𝑑𝑥^𝑖. (2.38) The number 𝑎_𝑖𝑗 becomes𝑎_𝑖𝑗=(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^𝑖−𝑗!𝑑𝑗!𝑖−𝑗^𝑖𝑓(𝑥)𝑑𝑥^𝑖_𝑥=𝑎2. (2.39)

(1) Coefficients with Two Indices
We write the numbers 𝑎_{𝑖−𝑙−𝑘,𝑗−𝑙} substituting the numbers 𝑎_𝑖𝑗 for numbers 𝑎^{𝑖+𝑗−1}𝑎_𝑖𝑗.
By formula (2.39), the following formula holds: 𝑎^{𝑖+𝑗−𝑘−2𝑙−1}𝑎_{𝑖−𝑙−𝑘,𝑗−𝑙}=(−1)^{𝑖−𝑘−𝑗}2^{𝑖−𝑘−𝑗}𝑎^{2𝑖−2𝑘−2𝑙−1}!!𝑑𝑗−𝑙𝑖−𝑘−𝑗^{𝑖−𝑘−𝑙}𝑓(𝑥)𝑑𝑥^{𝑖−𝑘−𝑙}_𝑥=𝑎2. (2.40) Let us substitute in (2.28): 𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}=^𝑖−𝑗_𝑘=0(−1)^{𝑖−𝑘−𝑗}2^{𝑖−𝑘−𝑗}𝑎^{2𝑖−2𝑘−2𝑙−1}!!𝑑𝑗−𝑙𝑖−𝑘−𝑗^{𝑖−𝑘−𝑙}𝑓(𝑥)𝑑𝑥^{𝑖−𝑘−𝑙}_𝑥=𝑎2(−1)^𝑘2^𝑘𝑥^{𝑙+𝑘+1/2}𝑑𝑙!𝑘!𝑑𝑥^{𝑙+𝑘𝑙+𝑘}1√𝑥. (2.41) Let 𝑥=𝑎². (2.42) Then, the following formula holds: 𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}=^𝑖−𝑗_𝑘=0(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^2𝑖!𝑑𝑗−𝑙𝑖−𝑘−𝑗!𝑙!𝑘!^{𝑖−𝑘−𝑙}𝑓(𝑥)𝑑𝑥^{𝑖−𝑘−𝑙}_𝑥=𝑎2𝑑^𝑙+𝑘𝑑𝑥^𝑙+𝑘1√𝑥_𝑥=𝑎2. (2.43) The sum equals 𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}=(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^2𝑖𝑑(𝑖−𝑗)!𝑗−𝑙!𝑙!𝑑𝑥^{𝑖−𝑗𝑖−𝑗}𝑑^𝑗−𝑙𝑓(𝑥)𝑑𝑥^𝑗−𝑙𝑑^𝑙𝑑𝑥^𝑙1√𝑥_𝑥=𝑎2. (2.44) Let us add with respect of index 𝑙 from zero to 𝑗: ^𝑗_𝑙=0𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}=^𝑗_𝑙=0(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^2𝑖𝑑(𝑖−𝑗)!𝑗−𝑙!𝑙!𝑑𝑥^{𝑖−𝑗𝑖−𝑗}𝑑^𝑗−𝑙𝑓(𝑥)𝑑𝑥^𝑗−𝑙𝑑^𝑙𝑑𝑥^𝑙1√𝑥_𝑥=𝑎2=(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^2𝑖𝑑(𝑖−𝑗)!𝑑𝑥^{𝑖−𝑗𝑗𝑖−𝑗}_𝑙=01𝑑𝑗−𝑙!𝑙!^𝑗−𝑙𝑓(𝑥)𝑑𝑥^𝑗−𝑙𝑑^𝑙𝑑𝑥^𝑙1√𝑥_𝑥=𝑎2. (2.45) The sum equals ^𝑗_𝑙=01𝑑𝑗−𝑙!𝑙!^𝑗−𝑙𝑓(𝑥)𝑑𝑥^𝑗−𝑙𝑑^𝑙𝑑𝑥^𝑙1√𝑥=1𝑑𝑗!𝑑𝑥^𝑗𝑗𝑓(𝑥)√𝑥. (2.46) We will substitute this sum in (2.45): ^𝑗_𝑙=0𝑎_{𝑖+𝑙,𝑗−𝑙,𝑙}=(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^2𝑖𝑑(𝑖−𝑗)!𝑑𝑥^{𝑖−𝑗𝑖−𝑗}1𝑑𝑗!𝑑𝑥^𝑗𝑗𝑓(𝑥)√𝑥_𝑥=𝑎2=(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^2𝑖𝑑(𝑖−𝑗)!𝑗!𝑑𝑥^𝑖𝑖𝑓(𝑥)√𝑥_𝑥=𝑎2. (2.47) We will substitute this sum in (2.19): 𝐴_𝑖𝑗=(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^2𝑖𝑑𝑗!(𝑖−𝑗)!𝑑𝑥^𝑖𝑖𝑓(𝑥)√𝑥_𝑥=𝑎2. (2.48)

2.2.2. A Series of Half-Integer Powers

Divide both sides of (2.33) by √𝑎²−2𝑎𝑏𝑐+𝑐²:∑^∞_𝑛=0𝑎_𝑛𝑎²−2𝑎𝑏𝑐+𝑐^2𝑛√𝑎²−2𝑎𝑏𝑐+𝑐²=∑^∞_𝑖=0∑^𝑖_𝑗=0𝑎_𝑖𝑗𝑏^𝑖−𝑗𝑐^𝑖+𝑗√𝑎²−2𝑎𝑏𝑐+𝑐²=∑^∞_𝑖=0∑^𝑖_𝑗=0𝑎_𝑖𝑗𝑏^𝑖−𝑗𝑐^𝑖+𝑗𝑎1−2(𝑏𝑐/𝑎)+𝑐²/𝑎². (2.49) By formula (2.21), the following formula holds:∑^∞_𝑖=0∑^𝑖_𝑗=0𝑎_𝑖𝑗𝑏^𝑖−𝑗𝑐^𝑖+𝑗𝑎1−2(𝑏𝑐/𝑎)+𝑐²/𝑎²=^∞𝑖_𝑖=0𝑗=0𝐴_𝑖𝑗𝑐^𝑖+𝑗𝑏^𝑖−𝑗, (2.50) where (i)coefficients 𝑎_𝑖𝑗 in (2.21) are replaced by coefficients 𝑎^{𝑖+𝑗−1}𝑎_𝑖𝑗, (ii)coefficients 𝐴_𝑖𝑗 are given by formula (2.48).

The left side of (2.49) becomes∑^∞_𝑛=0𝑎_𝑛𝑎²−2𝑎𝑏𝑐+𝑐^2𝑛√𝑎²−2𝑎𝑏𝑐+𝑐²=^∞𝑖_𝑖=0𝑗=0𝐴_𝑖𝑗𝑐^𝑖+𝑗𝑏^𝑖−𝑗. (2.51)

2.3. A Series of Half-Integer Powers

Consider the series ∑^∞_𝑛=0𝑎_𝑛𝑥^𝑛/2.

Let us separate this sum into (i)a sum with respect to even indices, (ii)a sum over odd indices.

Then, one has^∞_𝑛=0𝑎_𝑛𝑥^𝑛/2=^∞_𝑛=0𝑎_2𝑛𝑥^𝑛+^∞_𝑛=0𝑎_1+2𝑛𝑥^1/2+𝑛=^∞_𝑛=0𝑎_2𝑛𝑥^𝑛+∑^∞_𝑛=0𝑎_1+2𝑛𝑥^𝑛+1√𝑥. (2.52) In the second sum, we change the index 𝑛 for index 𝑛′=𝑛+1:^∞_𝑛=0𝑎_1+2𝑛𝑥^𝑛+1=^∞_𝑛′=1𝑎_{−1+2𝑛′}𝑥^𝑛′. (2.53) Define𝑎₋₁=0. (2.54) Then, the following formula holds:^∞_𝑛′=1𝑎_{−1+2𝑛′}𝑥^𝑛′=^∞_𝑛=0𝑎_−1+2𝑛𝑥^𝑛. (2.55) Formula (2.52) becomes^∞_𝑛=0𝑎_𝑛𝑥^𝑛/2=^∞_𝑛=0𝑎_2𝑛𝑥^𝑛+∑^∞_𝑛=0𝑎_−1+2𝑛𝑥^𝑛√𝑥. (2.56) We will denote 𝑓(𝑥)=^∞_𝑛=0𝑎_2𝑛𝑥^𝑛,𝑔(𝑥)=∑^∞_𝑛=0𝑎_−1+2𝑛𝑥^𝑛√𝑥. (2.57) Let𝑥=𝑎²−2𝑎𝑏𝑐+𝑐². (2.58) By formulas (2.33) and (2.51), the following formula holds:^∞_𝑛=0𝑎_𝑛𝑎²−2𝑎𝑏𝑐+𝑐^2𝑛/2=^∞𝑖_𝑖=0𝑗=0𝑎_𝑖𝑗𝑏^𝑖−𝑗𝑐^𝑖+𝑗+^∞𝑖_𝑖=0𝑗=0𝑏_𝑖𝑗𝑏^𝑖−𝑗𝑐^𝑖+𝑗, (2.59) where (see (2.39) and (2.48)) 𝑎_𝑖𝑗=(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^𝑖−𝑗!𝑑𝑗!𝑖−𝑗^𝑖𝑓(𝑥)𝑑𝑥^𝑖_𝑥=𝑎2,𝑏_𝑖𝑗=(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^2𝑖𝑑𝑗!(𝑖−𝑗)!^𝑖𝑔(𝑥)𝑑𝑥^𝑖_𝑥=𝑎2. (2.60) We will denote𝐴_𝑖𝑗=(−1)^𝑖−𝑗2^𝑖−𝑗𝑎^𝑖−𝑗!𝑑𝑗!𝑖−𝑗^𝑖𝑓(𝑥)𝑑𝑥^𝑖_𝑥=𝑎2+𝑎^𝑖+𝑗𝑑^𝑖𝑔(𝑥)𝑑𝑥^𝑖_𝑥=𝑎2. (2.61) Formula (2.59) becomes^∞_𝑛=0𝑎_𝑛𝑎²−2𝑎𝑏𝑐+𝑐^2𝑛/2=^∞𝑖_𝑖=0𝑗=0𝐴_𝑖𝑗𝑏^𝑖−𝑗𝑐^𝑖+𝑗. (2.62)

2.3.1. The Case of the Law of Cosines

Consider a number 𝜃.

Let𝑏=cos𝜃. (2.63)

The following formula appears in [2]:(cos𝜃)^𝑛=2^1−𝑛𝑛!^𝑛/2_𝑘=0cos((𝑛−2𝑘)𝜃)()!((.𝑘!𝑛−𝑘1+𝛿𝑛,2𝑘)) (2.64) Let us use this formula to rewrite formula (2.62): ^∞_𝑛=0𝑎_𝑛𝑎²−2𝑎𝑐cos𝜃+𝑐^2𝑛/2=^∞𝑖_𝑖=0𝑗=0𝐴_𝑖𝑗2^{1−𝑖+𝑗}𝑖−𝑗!^{𝑖/2−𝑗/2}_𝑘=0cos𝑖−𝑗−2𝑘𝜃!𝑐𝑘!𝑖−𝑗−𝑘1+𝛿𝑖−𝑗,2𝑘^𝑖+𝑗. (2.65) By Lemma A.2 (see the appendix), the following formula holds:^∞𝑖_𝑖=0𝑗=0𝐴_𝑖𝑗2^{1−𝑖+𝑗}𝑖−𝑗!^{𝑖/2−𝑗/2}_𝑘=0cos𝑖−𝑗−2𝑘𝜃!𝑐𝑘!𝑖−𝑗−𝑘1+𝛿𝑖−𝑗,2𝑘^𝑖+𝑗=^∞𝑖_𝑖=0^𝑗_𝑗=0𝑘=02^{1−𝑖+𝑗}𝐴_{𝑖+𝑘,𝑗−𝑘}4^−𝑘𝑖+2𝑘−𝑗!cos𝑖−𝑗𝜃𝑐^𝑖+𝑗!.𝑘!𝑖+𝑘−𝑗1+𝛿𝑖,𝑗 (2.66) By formula (2.61), the following formula holds:𝐴_{𝑖+𝑘,𝑗−𝑘}=(−1)^𝑖−𝑗2^{𝑖+2𝑘−𝑗}𝑎^{𝑖+2𝑘−𝑗}𝑑^𝑖+𝑘𝑓(𝑥)/𝑑𝑥^𝑖+𝑘_𝑥=𝑎2+𝑎^𝑖+𝑗(𝑑^𝑖+𝑘𝑔(𝑥)/𝑑𝑥^𝑖+𝑘)_𝑥=𝑎2!!.𝑗−𝑘𝑖+2𝑘−𝑗 (2.67) Multiply by 2^{1−𝑖+𝑗}4^−𝑘(𝑖+2𝑘−𝑗)!/𝑘!(𝑖+𝑘−𝑗)!(1+𝛿(𝑖,𝑗)):2^{1−𝑖+𝑗}𝐴_{𝑖+𝑘,𝑗−𝑘}4^−𝑘𝑖+2𝑘−𝑗!!()𝑘!𝑖+𝑘−𝑗1+𝛿𝑖,𝑗=2−1^𝑖−𝑗𝑎^{𝑖+2𝑘−𝑗}(𝑑^𝑖+𝑘𝑓(𝑥)/𝑑𝑥^𝑖+𝑘)_𝑥=𝑎2+𝑎^2𝑖+2𝑘(𝑑^𝑖+𝑘𝑔(𝑥)/𝑑𝑥^𝑖+𝑘)_𝑥=𝑎2!.𝑗−𝑘!𝑘!𝑖+𝑘−𝑗1+𝛿𝑖,𝑗 (2.68) Let us use formula (2.34) to rewrite2(−1)^𝑖−𝑗𝑎^{𝑖+2𝑘−𝑗}𝑑^𝑖+𝑘𝑓(𝑥)/𝑑𝑥^𝑖+𝑘_𝑥=𝑎2+𝑎^2𝑖+2𝑘𝑑^𝑖+𝑘𝑔(𝑥)/𝑑𝑥^𝑖+𝑘_𝑥=𝑎2!()𝑗−𝑘!𝑘!𝑖+𝑘−𝑗1+𝛿𝑖,𝑗=2−1^𝑖−𝑗𝑑^𝑗−𝑘𝑥^𝑖/𝑑𝑥^𝑗−𝑘_𝑥=𝑎2𝑎^−𝑖+𝑗𝑑^𝑖+𝑘𝑓(𝑥)/𝑑𝑥^𝑖+𝑘_𝑥=𝑎2+𝑎^2𝑗𝑑^𝑖+𝑘𝑔(𝑥)/𝑑𝑥^𝑖+𝑘_𝑥=𝑎2.𝑗−𝑘!𝑘!1+𝛿𝑖,𝑗𝑖! (2.69) Let us add with respect to the index 𝑘 from zero to 𝑗:^𝑗_𝑘=02(−1)^𝑖−𝑗(𝑑^𝑗−𝑘𝑥^𝑖/𝑑𝑥^𝑗−𝑘)_𝑥=𝑎2𝑎^−𝑖+𝑗(𝑑^𝑖+𝑘𝑓(𝑥)/𝑑𝑥^𝑖+𝑘)_𝑥=𝑎2+𝑎^2𝑗𝑑^𝑖+𝑘𝑔(𝑥)/𝑑𝑥^𝑖+𝑘)_𝑥=𝑎2()𝑗−𝑘!𝑘!1+𝛿𝑖,𝑗𝑖!=2−1^𝑖−𝑗𝑎^−𝑖+𝑗(𝑑^𝑗/𝑑𝑥^𝑗)𝑥^𝑖(𝑑^𝑖𝑓(𝑥)/𝑑𝑥^𝑖)_𝑥=𝑎2+𝑎^2𝑗𝑑^𝑗/𝑑𝑥^𝑗𝑥^𝑖𝑑^𝑖𝑔(𝑥)/𝑑𝑥^𝑖_𝑥=𝑎2.𝑖!𝑗!1+𝛿𝑖,𝑗 (2.70) Multiply by cos((𝑖−𝑗)𝜃)𝑐^𝑖+𝑗.

Let us add with respect to the index 𝑗 from zero to 𝑖.

Let us add with respect to the index 𝑖 from zero to infinity.

By Lemma A.1 (see the appendix), the following formula holds:^∞𝑖_𝑖=0𝑗=02(−1)^𝑖−𝑗𝑎^2𝑗(𝑑^𝑗/𝑑𝑥^𝑗)𝑎^{−𝑖−𝑗}𝑥^𝑖(𝑑^𝑖𝑓(𝑥)/𝑑𝑥^𝑖)+𝑥^𝑖(𝑑^𝑖𝑔(𝑥)/𝑑𝑥^𝑖)_𝑥=𝑎2𝜃𝑐𝑖!𝑗!1+𝛿𝑖,𝑗cos𝑖−𝑗^𝑖+𝑗=^∞∞_𝑗=0𝑖=02(−1)^𝑗𝑑^𝑖/𝑑𝑥^𝑖𝑥^𝑖+𝑗𝑑^𝑖+𝑗/𝑑𝑥^𝑖+𝑗𝑎^−𝑗𝑓(𝑥)+𝑎^2𝑖𝑔(𝑥)cos𝑗𝜃𝑐^2𝑖+𝑗!.𝑖!𝑖+𝑗1+𝛿0,𝑗 (2.71) We will denote 𝐵_𝑖𝑗()=2−1^𝑗𝑑^𝑖/𝑑𝑥^𝑖𝑥^𝑖+𝑗𝑑^𝑖+𝑗/𝑑𝑥^𝑖+𝑗𝑎^−𝑗𝑓(𝑥)+𝑎^2𝑖𝑔(𝑥)_𝑥=𝑎2!.𝑖!𝑖+𝑗1+𝛿0,𝑗 (2.72) The left side of (2.65) becomes^∞_𝑛=0𝑎_𝑛𝑎²−2𝑎𝑐cos𝜃+𝑐^2𝑛/2=^∞∞_𝑗=0𝑖=0𝐵_𝑖𝑗cos𝑗𝜃𝑐^2𝑖+𝑗. (2.73)

(1) A Formula for Calculating Coefficients
Consider two natural numbers 𝑚 and 𝑛.
Let 𝑓(𝑥)=𝑥^𝑚+𝑛,𝑔(𝑥)=0. (2.74) Formula (2.72) becomes 𝐵_𝑖𝑗()=2−1^𝑗𝑎^{−𝑗+2𝑚+2𝑛−2𝑖}((𝑚+𝑛)!)²()!!.𝑚+𝑛−𝑖𝑚+𝑛−𝑖−𝑗!𝑖!𝑖+𝑗1+𝛿0,𝑗 (2.75)

3. Functions on the Plane Expressed as a Linear Combination of Powers of Distances to Points

We deduce an identity regarding linear combinations of powers of distances from certain points, as well as similar new identities, from the formula of a power of the cosine. The formula of a power of the cosine is a powerful tool in mathematical analysis. In the space of analytic functions, we construct sums ∑^𝑁_𝑖=1𝑎_𝑖|𝐩−𝐩_𝑖|^𝑝 associated to them. We allow ourselves to use the name sum even if it is not finite.

Consider (i)two numbers 𝑟 and 𝜃, (ii)two natural numbers 𝑚 and 𝑛.

Assume that there are (i) numbers 𝜃₁,𝜃₂,…, (ii) positive numbers 𝑝₁,𝑝₂,…, and (iii) series ∑^∞_𝑖=0𝑎_𝑘𝑖(𝑝²_𝑘−2𝑝_𝑘𝑟cos(𝜃−𝜃_𝑘)+𝑟²)^𝑖/2, 𝑘=1,2,…, such that𝑟^𝑛+2𝑚cos(𝑛𝜃)=_𝑘^∞_𝑖=0𝑎_𝑘𝑖𝑝²_𝑘−2𝑝_𝑘𝑟cos𝜃−𝜃_𝑘+𝑟^2𝑖/2. (3.1) We will denote 𝑓_𝑘(𝑥)=^∞_𝑖=0𝑎_𝑘,2𝑖𝑥^𝑖,𝑔_𝑘(𝑥)=∑^∞_𝑖=0𝑎_{𝑘,−1+2𝑖}𝑥^𝑖√𝑥. (3.2) By formula (2.73), the following formula holds:^∞_𝑖=0𝑎_𝑘𝑖𝑝²_𝑘−2𝑝_𝑘𝑟cos𝜃−𝜃_𝑘+𝑟^2𝑖/2=^∞∞_𝑗=0𝑖=0𝐵_𝑘𝑖𝑗cos𝑗𝜃−𝜃_𝑘𝑟^2𝑖+𝑗, (3.3) where the coefficients 𝐵_𝑘𝑖𝑗 are given by formula (2.72): 𝐵_𝑖𝑗𝑘()=2−1^𝑗𝑑^𝑖/𝑑𝑥^𝑖𝑥^𝑖+𝑗(𝑑^𝑖+𝑗/𝑑𝑥^𝑖+𝑗)𝑎^−𝑗𝑓_𝑘(𝑥)+𝑎^2𝑖𝑔_𝑘(𝑥)_𝑥=𝑎2!.𝑖!𝑖+𝑗1+𝛿0,𝑗 (3.4) We use the formulacos𝜃−𝜙=cos(𝜃)cos𝜙+sin(𝜃)sin𝜙. (3.5) Formula (3.1) becomes𝑟^𝑛+2𝑚cos(𝑛𝜃)=_𝑘^∞∞_𝑗=0𝑖=0𝐵_𝑘𝑖𝑗cos𝑗𝜃cos𝑗𝜃_𝑘+sin𝑗𝜃sin𝑗𝜃_𝑘𝑟^2𝑖+𝑗. (3.6) The following formulas are sufficient conditions for the solution of the equation above:𝜃_𝑘=𝑘𝜋𝑛,𝑘=1,2,…,_𝑘cos𝑗𝑘𝜋𝑛𝐵_𝑘𝑖𝑗=𝛿(𝑖,𝑚)𝛿𝑗,𝑛_𝑘,sin𝑗𝑘𝜋𝑛𝐵_𝑘𝑖𝑗=0,𝑖,𝑗=0,1,…. (3.7)

3.1. The Product of a Power of the Distance to the Origin and Cosine

Consider (i)1+𝑚 numbers 𝑎₀,…,𝑎_𝑚(ii)1+𝑚 positive numbers 𝑝₀,…,𝑝_𝑚.

We will denote 𝑓_𝑘𝑙(𝑥)=(−1)^𝑘𝑎_𝑙𝑥^𝑚+𝑛,𝑘=1,…,2𝑛;𝑙=0,…,𝑚. (3.8) In Section 2.3.1(1), we computed the numbers 𝐵_𝑖𝑗. Let us denote 𝐵_{𝑘𝑙,𝑖𝑗}()=2−1^𝑘+𝑗𝑎_𝑙𝑝_𝑙^{−𝑗+2𝑚+2𝑛−2𝑖}((𝑚+𝑛)!)²()!!.𝑚+𝑛−𝑖𝑚+𝑛−𝑖−𝑗!𝑖!𝑖+𝑗1+𝛿0,𝑗 (3.9) Multiply by cos(𝑗𝑘𝜋/𝑛)(sin(𝑗𝑘𝜋/𝑛), resp.), and add up over the index 𝑘 from one to 2𝑛, ∑^2𝑛_𝑘=1𝐵_{𝑘𝑙,𝑖𝑗}cos(𝑗𝑘𝜋/𝑛) (∑^2𝑛_𝑙=1𝐵_{𝑘𝑙,𝑖𝑗}sin(𝑗𝑘𝜋/𝑛), resp.).

We use the formula^2𝑛_𝑘=1(−1)^𝑘𝑒^{√−1𝑗𝑘𝜋/𝑛}=2𝑛𝛿𝑗,𝑛. (3.10) The sums ∑^2𝑛_𝑘=1𝐵_{𝑘𝑙,𝑖𝑗}cos(𝑗𝑘𝜋/𝑛) and ∑^2𝑛_𝑘=1𝐵_{𝑘𝑙,𝑖𝑗}sin(𝑗𝑘𝜋/𝑛) are equal to^2𝑛_𝑘=1𝐵_{𝑘𝑙,𝑖𝑗}cos𝑗𝑘𝜋𝑛=4𝑛𝛿𝑗,𝑛(−1)^𝑗𝑎_𝑙𝑝_𝑙^{−𝑗+2𝑚+2𝑛−2𝑖}((𝑚+𝑛)!)²()!!=()𝑚+𝑛−𝑖𝑚+𝑛−𝑖−𝑗!𝑖!𝑖+𝑗1+𝛿0,𝑗4𝑛𝛿𝑗,𝑛−1^𝑛𝑎_𝑙𝑝_𝑙^{𝑛+2𝑚−2𝑖}((𝑚+𝑛)!)²()!()()!((,𝑚+𝑛−𝑖𝑚−𝑖!𝑖!𝑖+𝑛1+𝛿0,𝑛))_2𝑛𝑘=1𝐵_{𝑘𝑙,𝑖𝑗}sin𝑗𝑘𝜋𝑛=0. (3.11) We want the numbers 𝑎_𝑙 to satisfy the following equation:^𝑚_𝑙=0^2𝑛_𝑘=1𝐵_{𝑘𝑙,𝑖𝑗}cos𝑗𝑘𝜋𝑛=𝛿(𝑖,𝑚)𝛿𝑗,𝑛,𝑖=0,…,𝑚;𝑗=0,1,…. (3.12) By formula (3.11), this sum is reduced to^𝑚_𝑙=0𝑎_𝑙𝑝_𝑙^{𝑛+2𝑚−2𝑖}=𝛿(𝑖,𝑚)(1+𝛿(0,𝑛))(−1)^𝑛(𝑛−1)!𝑚!4(𝑚+𝑛)!,𝑖=0,…,𝑚. (3.13) This is a system of linear equations with unknowns 𝑎₀,…,𝑎_𝑚. The coefficient matrix for the system is 𝑝_𝑙^{𝑛+2𝑚−2𝑖}; 𝑖, 𝑙=0,…,𝑚.

Assume that the numbers 𝑝₀,…,𝑝_𝑚 are different from each other. Then,𝑎_𝑙=(1+𝛿(0,𝑛))(−1)^𝑛(𝑛−1)!𝑚!4()!𝑝𝑚+𝑛^𝑙−𝑛_𝑖≠𝑙𝑝²_𝑖𝑝²_𝑖−𝑝²_𝑙,𝑙=0,…,𝑚. (3.14) Equations (3.7) are valid. Therefore,𝑟^𝑛+2𝑚cos(𝑛𝜃)=^𝑚_𝑙=0^2𝑛_𝑘=1(−1)^𝑙𝑎_𝑙𝑝²_𝑙−2𝑝_𝑙𝑟cos𝜃−𝜃_𝑘+𝑟^2𝑚+𝑛, (3.15) where, as in (3.7),

𝜃_𝑘=𝑘𝜋𝑛,𝑘=1,2…. (3.16)

3.2. Example: The Difference between Two Paraboloids

Write(𝜃)=(𝜃)𝑟cos𝑟cos2+116+𝑟²−−𝑟cos(𝜃)2+116+𝑟². (3.17) In the coordinates (𝑥, 𝑦),𝑟cos(𝜃)=𝑥. (3.18)Figure 3 represents graphically Identity (3.17).

Figure 1 

Figure 2 

Figure 3 

3.3. Example: A Combination of Fourth-Degree Surfaces

Write𝑟²()1cos2𝜃=−8−1√2𝑟sin(𝜃)+𝑟²²+18+1√2𝑟cos(𝜃)+𝑟²²−18+1√2𝑟sin(𝜃)+𝑟²²+18−1√2𝑟cos(𝜃)+𝑟²². (3.19) In the coordinates (𝑥, 𝑦),𝑟²cos(2𝜃)=𝑥²−𝑦². (3.20)Figure 4 represents graphically Identity (3.19).

Figure 4 

Figure 5 

Representation of the difference 𝐩−𝐩𝑘𝑙.

Figure 6 

Graphics of both sides of formula (3.22), 𝑛=2,3. In the series on the right, we add up to 𝑚=6.

3.4. Example: The Product of a Bessel Function and Cosine

Consider the product 𝑟^𝑛∑^∞_𝑚=0(((−1)^𝑚/𝑚!(𝑛+𝑚)!)𝑟^2𝑚)cos𝑛𝜃.

We will denote 𝑎_𝑘𝑙𝑚=14(−1)^{𝑛+𝑘+𝑚}(1+𝛿(0,𝑛))𝑝_𝑙^−𝑛𝑛!((𝑚+𝑛)!)⁻²_𝑖≠𝑙𝑝_𝑖²𝑝_𝑖²−𝑝_𝑙². (3.21) By formula (3.15),𝑟^𝑛∞_𝑚=0(−1)^𝑚()!𝑟𝑚!𝑛+𝑚^2𝑚cos𝑛𝜃=^2𝑛∞_𝑘=1^𝑚_𝑚=0𝑙=0𝑎_𝑘𝑙𝑚𝑝²_𝑙−2𝑝_𝑙𝑟cos𝜃−𝜃_𝑘+𝑟^2𝑚+𝑛. (3.22) We will denote (),𝐩𝐩∶pointwithcoordinates𝑟cos𝜃,𝑟sin𝜃_𝑘𝑙∶pointwithcoordinates𝑝_𝑙cos𝜃_𝑘,𝑝_𝑙sin𝜃_𝑘,|𝐩|𝑠=𝑟,𝑦_𝑘𝑙(𝑥)=𝑥^𝑛∞_𝑚=𝑙𝑎_𝑘𝑙𝑚𝑥^𝑚. (3.23) Rewrite formula (3.22) as𝑟^𝑛∞_𝑚=0(−1)^𝑚()!𝑟𝑚!𝑛+𝑚^2𝑚cos𝑛𝜃=^2𝑛∞_𝑘=1𝑙=0𝑠_𝑘𝑙𝐩−𝐩_𝑘𝑙². (3.24)

Appendix

A. Two Lemmas

We give two formulas for changing the order of the sums in double series.

Lemma A.1. Consider the series ∑^∞_𝑖=0∑^𝑖_𝑗=0𝑎_{𝑖,𝑖−𝑗}. Assume that one can exchange the order of addition with respect to the indices 𝑖 and 𝑗. Then, the following formula holds: ^∞𝑖_𝑖=0𝑗=0𝑎_𝑖𝑗=^∞∞_𝑗=0𝑖=0𝑎_{𝑖+𝑗,𝑖}. (A.1)

Lemma A.2. Consider the series (i)∑^∞_𝑖=0∑^𝑖/2_𝑘=0∑^𝑖−2𝑘_𝑗=0𝑎_𝑖𝑗𝑘𝑦, (ii)∑^∞_𝑘=0∑^∞_𝑖=𝑘∑^𝑖_𝑗=𝑘𝑎_{𝑖+𝑘,𝑗−𝑘,𝑘}. Assume that one can exchange the order of addition with respect to the indices 𝑖 and 𝑘. Then, the following formula holds: ^∞𝑖_𝑖=0𝑗=0^{(𝑖−𝑗)/2}_𝑘=0𝑎_𝑖𝑗𝑘=^∞𝑖_𝑖=0^𝑗_𝑗=0𝑘=0𝑎_{𝑖+𝑘,𝑗−𝑘,𝑘}. (A.2)