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First I have this function:

$$g_0(1,eta)=fracfrac3etaeta_c-eta+sum_k=1^4kA_kleft(fracetaeta_cright)^k4eta$$

I need to multiply this function by another function:

Full polynomial function:

$$C00*r^0*rpf^0 + C10*r^1*rpf^0 + C20*r^2*rpf^0 + C30*r^3*rpf^0 + C40*r^4*rpf^0 + C50*r^5*rpf^0 + C60*r^6*rpf^0 + C01*r^0*rpf^1 + C11*r^1*rpf^1 + C21*r^2*rpf^1 + C31*r^3*rpf^1 + C41*r^4*rpf^1+ C51*r^5*rpf^1 + C61*r^6*rpf^1 + C02*r^0*rpf^2 + C12*r^1*rpf^2
+ C22*r^2*rpf^2 + C32*r^3*rpf^2 + C42*r^4*rpf^2 + C52*r^5*rpf^2 + C62*r^6*rpf^2 + C03*r^0*rpf^3 + C13*r^1*rpf^3 + C23*r^2*rpf^3 + C33*r^3*rpf^3 + C43*r^4*rpf^3 + C53*r^5*rpf^3 + C63*r^6*rpf^3+ C04*r^0*rpf^4 + C14*r^1*rpf^4 + C24*r^2*rpf^4 + C34*r^3*rpf^4 + C44*r^4*rpf^4 + C54*r^5*rpf^4 + C64*r^6*rpf^4 + C05*r^0*rpf^5 + C15*r^1*rpf^5 + C25*r^2*rpf^5 + C35*r^3*rpf^5 + C45*r^4*rpf^5
+ C55*r^5*rpf^5 + C65*r^6*rpf^5 + C06*r^0*rpf^6 + C16*r^1*rpf^6 +
+ C26*r^2*rpf^6 + C36*r^3*rpf^6 + C46*r^4*rpf^6 + C56*r^5*rpf^6 +
+ C66*r^6*rpf^6;$$

Note that rpf is just $eta/eta_c$

Then I need to accumulate all terms in a series expansion about $fracetaeta_c=0$. I need to collect the coefficients in front of each order polynomial. Can anyone help with this?

If you care only about the $(eta/eta_c)^1$ terms, you can safely ignore higher order terms in both expansions. From your previous question, you can write $g_0=a+b*rpf$. From your polynomial just keep terms of $rpf^0$ and $rpf^1$, so you are left with $c+d*rpf$. Here $c=C00*r^0 + C10*r^1 + C20*r^2 + C30*r^3 + C40*r^4 + C50*r^5 + C60*r^6$ and $d=C01*r^0+ C11*r^1 + C21*r^2 + C31*r^3 + C41*r^4+ C51*r^5 + C61*r^6$. With these, the product becomes $$ac+(ad+bc)*rpf+bd*rpf^2$$
And you can ignore the last term, since it's already higher power in $rpf$.

Since $rpf=fracetaeta_C$ the polynomial can be written as
beginalign*
sum_j=0^6sum_l=0^6C_j,lr^jleft(fracetaeta_Cright)^ltag1
endalign*
The function $g_0left(1,etaright)$ can be written assuming $left|fracetaeta_Cright|<1$ as
beginalign*
g_0left(1,etaright)&=fracfrac3etaeta_c-eta+sum_k=1^4kA_kleft(fracetaeta_cright)^k4eta\
&=frac34frac1eta_C-eta+frac14eta_Csum_k=1^4kA_kleft(fracetaeta_Cright)^k-1\
&=frac34eta_Csum_k=0^inftyleft(fracetaeta_Cright)^k+frac14eta_Csum_k=1^4kA_kleft(fracetaeta_Cright)^k-1tag2\
endalign*

According to a comment from OP the focus is the linear term of the product of the polynomial with $g_0$. It is convenient to use the coefficient of operator $[z^n]$ to denote the coefficient of $z^n$.

Let $z=fracetaeta_C$. We obtain from (1) and (2)
beginalign*
colorblue[z^1]&colorblueleft(frac34eta_Csum_k=0^infty z^k+frac14eta_Csum_k=0^3(k+1)A_k+1z^kright)
sum_j=0^6sum_l=0^6C_j,lr^jz^l\
&=frac34eta_Csum_k=0^1 [z^1-k]sum_j=0^6sum_l=0^6C_j,lr^jz^l
+frac14eta_Csum_k=0^1[z^1-k](k+1)A_k+1
sum_j=0^6sum_l=0^6C_j,lr^jz^ltag3\
&=frac34eta_Cleft([z^0]+[z^1]right)sum_j=0^6sum_l=0^6C_j,lr^jz^l
+frac14eta_Cleft(2A_2[z^0]+A_1[z^1]right)
sum_j=0^6sum_l=0^6C_j,lr^jz^l\
&=frac34eta_Csum_j=0^6left(C_j,0+C_j,1right)r^j
+frac14eta_Csum_j=0^6left(2A_2C_j,0+A_1C_j,1right)r^jtag4\
&,,colorblue=frac14eta_Csum_j=0^6left((3+2A_2)C_j,0+(3+A_1)C_j,1right)r^j
endalign*

Comment:

• In (3) we apply the rule $[z^p-q]A(z)=[z^p]z^qA(z)$. We restrict the upper index of the sums with $1$ since other terms do not contribute to $z^1$.




• In (4) we select the coefficients accordingly.