> #recurrence relation (2.7) on page 25
> rsolve( { r(0)=alpha, r(n)=r(n-1)+beta+gamma*n}, r );

      alpha + gamma (n + 1) (1/2 n + 1) + (- gamma + beta) (n + 1) - beta

> collect(",n);

                             2
                  1/2 gamma n  + (1/2 gamma + beta) n + alpha

> rsolve(a(n)*T(n)=b(n)*T(n-1)+c(n),T);

 /  n - 1              \ // n - 1                                    \       \
 | --------'           | || -----                                    |       |
 |'  |  |    b(_n1 + 1)| ||  \                 c(_n2 + 1)            |       |
 |   |  |    ----------| ||   )    ----------------------------------| + T(0)|
 |   |  |    a(_n1 + 1)| ||  /                /   _n2               \|       |
 |   |  |              | || -----             | --------'           ||       |
 \ _n1 = 0             / ||_n2 = 0            |'  |  |    b(_n1 + 1)||       |
                         ||        a(_n2 + 1) |   |  |    ----------||       |
                         ||                   |   |  |    a(_n1 + 1)||       |
                         ||                   |   |  |              ||       |
                         \\                   \ _n1 = 0             //       /

> rsolve(c(n)=n+1+2/n*sum(c(k),k=0..n-1),c);

                                             n - 1
                                             -----
                                              \
                                               )   c(k)
                                              /
                                             -----
                                             k = 0
                     rsolve(c(n) = n + 1 + 2 ----------, c)
                                                  n

> rsolve({c(0)=0,n*c(n)=(n+1)*c(n-1)+2*n},c);
bytes used=72533760, alloc=42328504, time=51.72
                       2 (Psi(n + 2) - 1 + gamma) (n + 1)