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Referring to the spherical harmonics expansion in this article:

Méléard, P., Pott, T., Bouvrais, H., & Ipsen, J. H. (2011). Advantages of statistical analysis of giant vesicle flickering for bending elasticity measurements. The European Physical Journal E, 34(10), 1-14. DOI 10.1140/epje/i2011-11116-6 (for those who can't access the DOI, here are some snapshots of the relevant pages https://i.stack.imgur.com/sfQ1C.png and https://i.stack.imgur.com/LelKb.png)

The author claims that you can rearrange the summation in the last term of one equation:

$\sum_{n\geq2,|m|\leq n}^{n_{max}}U^m_n(t)Y^m_n(\theta,\varphi)$ ; last term of eq (2), page 3

where $U^m_n$ are the time-dependent coefficients of a spherical harmonics $Y^m_n$ expansion, and $\theta$ and $\phi$ are the spherical angles; as:

$\sum_{n\geq2}^{n_{max}}a^0_n(t)P^m_n(cos\theta) + \sum_{0 < m\leq n}(a^m_n(t)cos(m\varphi)+b^m_n(t)sin(m\varphi))P^m_n(cos\theta)$; last term of eq (3), same page

by using the time-dependent functions $a^m_n(t)$ and $b^m_n(t)$, which they define as:

$U^m_n = \frac{1}{2}(a^m_n(t) - ib^m_n(t))$ with $i=\sqrt{-1}$

together with their expression for the spherical harmonics $Y^m_n = P^m_n(cos\theta)e^{im\varphi}$, with $P^m_n$ related to the associated Legendre function (last line of page 3 and first line of page 4).

However, I don't seem to get the same:

$\sum_{n\geq2,|m|\leq n}^{n_{max}}U^m_n(t)Y^m_n(\theta,\varphi)=\sum_{n\geq2,|m|\leq n}^{n_{max}}\frac{1}{2}(a^m_n(t)-ib^m_n(t))P^m_n(cos\theta)e^{im\varphi}$

$=\sum_{n\geq2,|m|\leq n}^{n_{max}}\frac{1}{2}(a^m_n(t)-ib^m_n(t))P^m_n(cos\theta)(cos(m\varphi)+isin(m\varphi))$

$=\sum_{n\geq2,|m|\leq n}^{n_{max}}\frac{1}{2}(a^m_n(t)cos(m\varphi)+ia^m_n(t)sin(m\varphi) - ib^m_n(t)cos(m\varphi)+b^m_n(t)sin(m\varphi))P^m_n(cos\theta)$

$=\sum_{n\geq2}^{n_{max}}\frac{1}{2}(a^0_n(t) - ib^0_n(t))P^0_n(cos\theta) + \sum_{|m|\leq n,m\ne0}\frac{1}{2}(a^m_n(t)cos(m\varphi)+ia^m_n(t)sin(m\varphi) - ib^m_n(t)cos(m\varphi)+b^m_n(t)sin(m\varphi))P^m_n(cos\theta)$

In equation (4), page 4 they write that:

$|U^0_n(t)|^2=|a^0_n(t)|^2$

and

$|U^m_n(t)|^2+|U^{-m}_n(t)|^2=\frac{1}{2}[|a^m_n(t)|^2+|b^m_n(t)|^2], 1\leq m \leq n $

which implies that $U^m_n$ and $U^{-m}_n$ are complex conjugates. This means that

$a^m_n=a^{-m}_n$,

$b^m_n=-b^{-m}_n$, and

$b^0_n=0$, so we have

$\sum_{n\geq2}^{n_{max}}\frac{1}{2}a^0_n(t)P^0_n(cos\theta) + \sum_{0< m\leq n}\frac{1}{2}(2a^m_n(t)cos(m\varphi)+ia^m_n(t)0)P^m_n(cos\theta) + \sum_{|m|\leq n,m\ne0}\frac{1}{2}(- ib^m_n(t)cos(m\varphi)+b^m_n(t)sin(m\varphi))P^m_n(cos\theta)$

since $cos(x)+cos(-x)=2cos(x)$ and $sin(x)+sin(-x)=0$, so I'm left with

$=\sum_{n\geq2}^{n_{max}}\frac{1}{2}a^0_n(t)P^0_n(cos\theta) + \sum_{0< m\leq n}a^m_n(t)cos(m\varphi)P^m_n(cos\theta)+\sum_{|m|\leq n,m\ne0}\frac{1}{2}(-ib^m_n(t)cos(m\varphi)+b^m_n(t)sin(m\varphi))P^m_n(cos\theta)$

which is not what is written in eq (3) of the article

Could somebody help me find out what is that I'm misssing here?

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