Polygamma Function - Reflection Relation

Reflection Relation

(-1)^m \psi^{(m)} (1-z) - \psi^{(m)} (z) = \pi \frac{d^m}{d z^m} \cot{(\pi z)} = \pi^{m+1} \frac{P_m(\cos(\pi z))}{\sin^{m+1}(\pi z)}

where is alternatingly an odd resp. even polynomial of degree with integer coefficients and leading coefficient . They obey the recursion equation with .

Read more about this topic:  Polygamma Function

Famous quotes containing the words reflection and/or relation:

    “The Americans ... have invented so wide a range of pithy and hackneyed phrases that they can carry on an amusing and animated conversation without giving a moment’s reflection to what they are saying and so leave their minds free to consider the more important matters of big business and fornication.”
    —W. Somerset Maugham (1874–1965)

    “The adolescent does not develop her identity and individuality by moving outside her family. She is not triggered by some magic unconscious dynamic whereby she rejects her family in favour of her peers or of a larger society.... She continues to develop in relation to her parents. Her mother continues to have more influence over her than either her father or her friends.”
    —Terri Apter (20th century)