In [2]: hyperexpand(hyper([1, a, a, a], [a + 1, a + 1, a + 1], z)/a**3) Out[2]: lerchphi(z, 3, a) In [3]: combsimp(hyperexpand(meijerg([0, 1-a, 1-a, 1-a], [], [0], [-a, -a, -a], ...: exp_polar(-I*pi)*z))) Out[3]: lerchphi(z, 3, a) In [10]: hyperexpand(hyper([1, 1, 1, S(1)/2, S(1)/2, S(1)/3], [4, 2, S(3)/2, S(-1)/2, S(4)/3], x)) Out[10]: -2⋅ⅈ⋅π ⎛ 2⋅ⅈ⋅π ⎞ -4⋅ⅈ⋅π ⎛ 4⋅ⅈ⋅π ⎞ ────── ⎜ ───── ⎟ ────── ⎜ ───── ⎟ ⎛ -ⅈ⋅π ⎞ ⎛ -ⅈ⋅π ⎞ ⎛ ⎽⎽⎽ -ⅈ⋅π ⎞ ⎛ ⎽⎽⎽ -ⅈ⋅π ⎞ ⎛3 ⎽⎽⎽ -ⅈ⋅π ⎞ 3 ⎜3 ⎽⎽⎽ 3 -ⅈ⋅π ⎟ 3 ⎜3 ⎽⎽⎽ 3 -ⅈ⋅π ⎟ 21 6⋅log⎝x⋅ℯ + 1⎠ 2⋅polylog(2, x) 9 9 9⋅log⎝x⋅ℯ + 1⎠ 64⋅log⎝- ╲╱ x ⋅ℯ + 1⎠ 64⋅log⎝╲╱ x ⋅ℯ + 1⎠ 81⋅log⎝╲╱ x ⋅ℯ + 1⎠ 81⋅ℯ ⋅log⎝╲╱ x ⋅ℯ ⋅ℯ + 1⎠ 81⋅ℯ ⋅log⎝╲╱ x ⋅ℯ ⋅ℯ + 1⎠ - ── - ────────────────── + ─────────────── - ─── - ──── - ────────────────── - ───────────────────────── + ─────────────────────── - ─────────────────────── - ────────────────────────────────────── - ────────────────────────────────────── 80 x x 4⋅x 2 3 ⎽⎽⎽ ⎽⎽⎽ 3 ⎽⎽⎽ 3 ⎽⎽⎽ 3 ⎽⎽⎽ 2⋅x 2⋅x 5⋅╲╱ x 5⋅╲╱ x 16⋅╲╱ x 16⋅╲╱ x 16⋅╲╱ x In [16]: summation(x**n/(n+a)**3, (n,0,oo)) Out[16]: ⎧ lerchphi(x, 3, a) for │x│ ≤ 1 ⎪ ⎪ ∞ ⎪____ ⎪\ ` ⎪ \ n ⎨ \ x ⎪ ) ───────────────────────── otherwise ⎪ / 3 2 2 3 ⎪ / a + 3⋅a ⋅n + 3⋅a⋅n + n ⎪/___, ⎪n = 0 ⎩ In [17]: summation(x**n/n**4, (n, 1, oo)) Out[17]: ⎧polylog(4, x) for │x│ ≤ 1 ⎪ ⎪ ∞ ⎪ ____ ⎪ \ ` ⎪ \ n ⎨ \ x ⎪ ) ── otherwise ⎪ / 4 ⎪ / n ⎪ /___, ⎪ n = 1 ⎩ In [24]: expand_func(summation(1/(n+a)**3, (n, 0, oo))) Out[24]: ζ(3, a) In [25]: summation(1/n**2, (n, 1, oo)) Out[25]: 2 π ── 6 In [26]: summation(1/n**3, (n, 1, oo)) Out[26]: ζ(3) In [27]: summation(1/n**4, (n, 1, oo)) Out[27]: 4 π ── 90