Multiplication theorem

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In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. Multiplication theorem_sentence_0

For the explicit case of the gamma function, the identity is a product of values; thus the name. Multiplication theorem_sentence_1

The various relations all stem from the same underlying principle; that is, the relation for one special function can be derived from that for the others, and is simply a manifestation of the same identity in different guises. Multiplication theorem_sentence_2

Finite characteristic Multiplication theorem_section_0

The multiplication theorem takes two common forms. Multiplication theorem_sentence_3

In the first case, a finite number of terms are added or multiplied to give the relation. Multiplication theorem_sentence_4

In the second case, an infinite number of terms are added or multiplied. Multiplication theorem_sentence_5

The finite form typically occurs only for the gamma and related functions, for which the identity follows from a p-adic relation over a finite field. Multiplication theorem_sentence_6

For example, the multiplication theorem for the gamma function follows from the Chowla–Selberg formula, which follows from the theory of complex multiplication. Multiplication theorem_sentence_7

The infinite sums are much more common, and follow from characteristic zero relations on the hypergeometric series. Multiplication theorem_sentence_8

The following tabulates the various appearances of the multiplication theorem for finite characteristic; the characteristic zero relations are given further down. Multiplication theorem_sentence_9

In all cases, n and k are non-negative integers. Multiplication theorem_sentence_10

For the special case of n = 2, the theorem is commonly referred to as the duplication formula. Multiplication theorem_sentence_11

Gamma function–Legendre formula Multiplication theorem_section_1

The duplication formula and the multiplication theorem for the gamma function are the prototypical examples. Multiplication theorem_sentence_12

The duplication formula for the gamma function is Multiplication theorem_sentence_13

It is also called the Legendre duplication formula or Legendre relation, in honor of Adrien-Marie Legendre. Multiplication theorem_sentence_14

The multiplication theorem is Multiplication theorem_sentence_15

for integer k ≥ 1, and is sometimes called Gauss's multiplication formula, in honour of Carl Friedrich Gauss. Multiplication theorem_sentence_16

The multiplication theorem for the gamma functions can be understood to be a special case, for the trivial Dirichlet character, of the Chowla–Selberg formula. Multiplication theorem_sentence_17

Polygamma function, harmonic numbers Multiplication theorem_section_2

The polygamma function is the logarithmic derivative of the gamma function, and thus, the multiplication theorem becomes additive, instead of multiplicative: Multiplication theorem_sentence_18

The polygamma identities can be used to obtain a multiplication theorem for harmonic numbers. Multiplication theorem_sentence_19

Hurwitz zeta function Multiplication theorem_section_3

For the Hurwitz zeta function generalizes the polygamma function to non-integer orders, and thus obeys a very similar multiplication theorem: Multiplication theorem_sentence_20

and Multiplication theorem_sentence_21

Multiplication formulas for the non-principal characters may be given in the form of Dirichlet L-functions. Multiplication theorem_sentence_22

Periodic zeta function Multiplication theorem_section_4

The periodic zeta function is sometimes defined as Multiplication theorem_sentence_23

where Lis(z) is the polylogarithm. Multiplication theorem_sentence_24

It obeys the duplication formula Multiplication theorem_sentence_25

As such, it is an eigenvector of the Bernoulli operator with eigenvalue 2. Multiplication theorem_sentence_26

The multiplication theorem is Multiplication theorem_sentence_27

The periodic zeta function occurs in the reflection formula for the Hurwitz zeta function, which is why the relation that it obeys, and the Hurwitz zeta relation, differ by the interchange of s → −s. Multiplication theorem_sentence_28

The Bernoulli polynomials may be obtained as a limiting case of the periodic zeta function, taking s to be an integer, and thus the multiplication theorem there can be derived from the above. Multiplication theorem_sentence_29

Similarly, substituting q = log z leads to the multiplication theorem for the polylogarithm. Multiplication theorem_sentence_30

Polylogarithm Multiplication theorem_section_5

The duplication formula takes the form Multiplication theorem_sentence_31

The general multiplication formula is in the form of a Gauss sum or discrete Fourier transform: Multiplication theorem_sentence_32

These identities follow from that on the periodic zeta function, taking z = log q. Multiplication theorem_sentence_33

Kummer's function Multiplication theorem_section_6

The duplication formula for Kummer's function is Multiplication theorem_sentence_34

and thus resembles that for the polylogarithm, but twisted by i. Multiplication theorem_sentence_35

Bernoulli polynomials Multiplication theorem_section_7

For the Bernoulli polynomials, the multiplication theorems were given by Joseph Ludwig Raabe in 1851: Multiplication theorem_sentence_36

and for the Euler polynomials, Multiplication theorem_sentence_37

and Multiplication theorem_sentence_38

The Bernoulli polynomials may be obtained as a special case of the Hurwitz zeta function, and thus the identities follow from there. Multiplication theorem_sentence_39

Bernoulli map Multiplication theorem_section_8

Perhaps not surprisingly, the eigenvectors of this operator are given by the Bernoulli polynomials. Multiplication theorem_sentence_40

That is, one has that Multiplication theorem_sentence_41

Assuming that the sum converges, so that g(x) exists, one then has that it obeys the multiplication theorem; that is, that Multiplication theorem_sentence_42

Characteristic zero Multiplication theorem_section_9

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: theorem.