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I am reading book concrete mathematics, author talks about general method of converting recurrence relation of type $$a_nT_n=b_nT_{n−1}+c_n$$

multiplies the above relations with $s_n$ called summation factor.

which makes the equation : $$s_na_nT_n=s_nb_nT_{n−1}+s_nc_n$$

such that $$s_nb_n=s_{n−1}a_{n−1}$$

making $S_n=s_na_nT_n$ so $$S_n=S_{n-1}+s_nc_n$$ what i didn't understand is he write that $$S_n=s_0a_0T_0+\sum_{k=1}^{n}{s_kc_k}=s_1a_1T_0+\sum_{k=1}^{n}{s_kc_k}$$ how?

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Just note that

$$ S_n - S_{n-1} =s_n c_n$$

Summing from $1$ to $n$ it gives :

$$S_n - S_0 = \sum_{k=0}^n s_k c_k $$

(see telescoping sums). Given that $S_0 = s_0a_0T_0$ and $s_1b_1 = s_0a_0$ you finally have :

$$S_n=s_0a_0T_0+\sum_{k=1}^{n}{s_kc_k}=s_1 b_1T_0+\sum_{k=1}^{n}{s_kc_k}$$

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