The first polygamma functions in real terms
m = 0 m = 1 m = 2 m = 3 m = 4
In mathematics, the polygamma functions are a series of special functions that are defined as the derivatives of the function . The gamma function and the natural logarithm denote .
![{\ displaystyle \ psi _ {n} (z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e96858afcc57bdfbab72447722c04d681b7c0a)
![{\ displaystyle \ ln \ Gamma (z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22f3c0c0516093aa04c131e4e241f073896f9ff0)
![\ Gamma (z)](https://wikimedia.org/api/rest_v1/media/math/render/svg/11ca17f880240539116aac7e6326909299e2a080)
![\ ln](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0de5ba4f372ede555d00035e70c50ed0b9625d0)
The first two polygamma functions are called the digamma function and the trigamma function .
Representation of the first five polygamma functions in the complex plane
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notation
The polygamma functions are identified with the small Greek letter Psi . In the case of the first polygamma function, the index is usually left out or defined as 0; it is called the digamma function . The second polygamma function, i.e. the trigamma function , has the symbol (or less often ) and is the second derivative of . In general, the is -th polygamma function or polygamma function of the order with or referred to and than the th derivative of defined.
![\ psi (z)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c02965f8dd8bfe2c0352b07c1193b8dc276c1d8)
![\ psi _ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfdde1da54e02a016fe2a230c58b25dfcc014d6)
![\ psi ^ {{(1)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f824d0b8ec54f716ece0f1d8acf4898cab2f7dd)
![{\ displaystyle \ ln \ Gamma (z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22f3c0c0516093aa04c131e4e241f073896f9ff0)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![\ psi _ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5ff71e5c69dbfebe7cd4530de07406eff55c7a)
![\ psi ^ {{(n)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3778fedf71601aed1b639d2c4bbe246359b42cbd)
![(n + 1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b30a29cfd35628469f9dbffea4804f5b422f3037)
![\ ln \ Gamma (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/504f5bc84c572fb46ef4d3c4e09b7e3452cb6cdf)
Definition and further representations
It is
![{\ displaystyle \ psi _ {m} (z) = {\ frac {\ mathrm {d} ^ {m + 1}} {\ mathrm {d} z ^ {m + 1}}} \ ln \ Gamma (z ) = {\ frac {\ mathrm {d} ^ {m}} {\ mathrm {d} z ^ {m}}} \, \ psi (z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4076376893006249b1058a09e9a716aed156db7)
with the digamma function . Such derivatives are also referred to as logarithmic derivatives of .
![\ psi (z)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c02965f8dd8bfe2c0352b07c1193b8dc276c1d8)
![{\ displaystyle \ Gamma (\ cdot)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb56054f7834e05862ff50930fc46b4f8e3f8280)
An integral representation is
![\ psi _ {m} (z) = (- 1) ^ {{m + 1}} \ int \ limits _ {0} ^ {\ infty} {\ frac {t ^ {m} {\ mathrm e} ^ {{-zt}}} {1 - {\ mathrm e} ^ {{- t}}}} \, {\ mathrm d} t](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1876d03a64b2a3dd030d622af9e0a306e9159e1)
for and![{\ displaystyle {\ rm {{Re} \, z> 0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cefe128d6d18869e3e4b44825a4b811ebf5818a2)
properties
Difference equations
The polygamma functions have the difference equations
![\ psi _ {m} (z + 1) = \ psi _ {m} (z) + (- 1) ^ {m} \; m! \; z ^ {{- m-1}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/60a49e12f5ddff04013276cb1af7c8f745753c42)
Reflection formula
Another important relationship is
![(-1) ^ {m} \ psi _ {m} (1-z) - \ psi _ {m} (z) = \ pi {\ frac {{\ mathrm d} ^ {m}} {{\ mathrm d} z ^ {m}}} \ cot {(\ pi z)}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9740c3fb0714603a11158ed063afad191aba0be)
Multiplication formula
The multiplication formula is given for by
![m> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/501173910e6da8425b4e9d44a4e8643620bc2464)
![\ sum _ {{k = 0}} ^ {{n-1}} \ psi _ {{m}} \ left ({\ frac {z + k} {n}} \ right) = n ^ {{m +1}} \ psi _ {{m}} (z).](https://wikimedia.org/api/rest_v1/media/math/render/svg/7241ab3d881242e4eebac89fcb82c6b701e203bc)
For the case of the digamma function , see there.
![m = 0,](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9da3557183c12da744b4aa38751653a795f68c)
Series representations
A series representation of the polygamma function is
![\ psi _ {m} (z) = (- 1) ^ {{m + 1}} \; m! \; \ sum _ {{k = 0}} ^ {\ infty} {\ frac 1 {(z + k) ^ {{m + 1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4081c670a2b5e9f670f39597487be554fe01b058)
where and is any complex number other than negative integers. The formula is easier to write using Hurwitz's zeta function than
![m> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/501173910e6da8425b4e9d44a4e8643620bc2464)
![\ zeta (x, y)](https://wikimedia.org/api/rest_v1/media/math/render/svg/60731acbf569690a4496af3cbbd858c38e1a0b37)
![\ psi _ {m} (z) = (- 1) ^ {{m + 1}} \; m! \; \ zeta (m + 1, z).](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3707a14fe5eb18f8770d35a82d493391a8db81)
The generalization of the polygamma functions to arbitrary, non-whole orders is given below .
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
Another series representation is
![\ psi _ {m} (z) = - \ gamma \ delta _ {{m, 0}} \; - \; {\ frac {(-1) ^ {m} m!} {z ^ {{m + 1}}}} \; + \; \ sum _ {{k = 1}} ^ {{\ infty}} \ left ({\ frac {1} {k}} \ delta _ {{m, 0}} \; - \; {\ frac {(-1) ^ {m} m!} {(z + k) ^ {{m + 1}}}} \ right),](https://wikimedia.org/api/rest_v1/media/math/render/svg/765a490d5ea3c9c54cede0c835b4e04378e05900)
where the Kronecker delta denotes, which follows from the decomposition of the gamma function according to Weierstrass' product theorem.
![\ delta _ {{n, 0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/311fdd8b186a70502e3271e27a91d37b57610ff8)
The Taylor series um is given by
![z = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/078535cde78d90bfa1d9fbb2446204593a921d57)
![\ psi _ {m} (z + 1) = \ sum _ {{k = 0}} ^ {\ infty} (- 1) ^ {{m + k + 1}} (m + k)! \; \ zeta (m + k + 1) \; {\ frac {z ^ {k}} {k!}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/25d5372db57bf707d6b0a199d086eb807c0f625c)
which converges for. referred to the Riemann zeta function .
![| z | <1](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c0fa57b899b653a3823f85f43fd666309c09b3)
![\ zeta](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae)
Special values
The values of the Polygammafunktionen for rational arguments can usually be expressed using constants and functions such as , square root , Clausen function , Riemann ζ function , Catalan's constant and Dirichlet β function ; z. B.
![\pi](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a)
![{\ mathrm {Cl}} (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/872d0f0db8c3a914b5d795f4892eb9d546418f85)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![\ psi _ {m} ({\ tfrac 12}) = (- 1) ^ {{m + 1}} m! \, (2 ^ {{{m + 1}} - 1) \ zeta (m + 1) , \ qquad m \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/553811856c83c6050d3d4e2c41c26912cc528f40)
In general, the following also applies:
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The mth derivative of the tangent can also be expressed using the polygamma function:
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In addition, special values of polygamma functions have repeatedly proven useful as universal constants in a closed limit value description of series or integrals, for example applies
![{\ displaystyle \ sum _ {n = 0} ^ {\ infty} {\ frac {(-1) ^ {n}} {(2n + 1) ^ {4}}} = {\ frac {1} {768 }} \ left (\ psi _ {3} \ left ({\ tfrac {1} {4}} \ right) -8 \ pi ^ {2} \ right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b5f28616cfe81cb0b50ddac77824b1b0fa4d96b)
Generalized polygamma function
The generalized polygamma function satisfies for and the functional equation
![{\ displaystyle s \ in \ mathbb {C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f13da7c2bac9dd6324ac83093fb42f3a9e86fbd0)
![{\ displaystyle z \ in \ mathbb {C} \ setminus - \ mathbb {N} _ {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b73d1e7b2cadd04b90f3efc42cacf9d80413f1)
![\ psi _ {s} (z + 1) = \ psi _ {s} (z) + {\ frac {\ ln z- \ psi (-s) - \ gamma} {\ Gamma (-s)}} \ , z ^ {{- (s + 1)}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/efb3f5d28eef9542b38e7086d007e2598af827e9)
where denotes the Euler-Mascheroni constant . Because of
![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
![{\ frac {\ psi (-m)} {\ Gamma (-n)}} = (- 1) ^ {{n-1}} n!](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bcd6d6b98483952b22fc91e3d4dcab0a52e1d45)
for integers , the difference equation given above for natural is included.
![m, n \ geq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ea6a401de3d862453e87db5876a99ead9647d81)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
The relationship is then obtained
with the aid of the Hurwitz function![\ zeta](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae)
![\ psi _ {s} (z) = {\ frac 1 {\ Gamma (-s)}} \ left ({\ frac {\ partial} {\ partial s}} + \ psi (-s) + \ gamma \ right) \ zeta (s + 1, z) = {\ mathrm e} ^ {{- \ gamma \, s}} {\ frac {\ partial} {\ partial s}} \ left ({\ mathrm e} ^ {{\ gamma \, s}} \, {\ frac {\ zeta (s + 1, z)} {\ Gamma (-s)}} \ right),](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e2280833dc61dced125a6a6d274a5c9cd1bcd3)
which satisfies the functional equation.
As a consequence, the doubling formula
![\ psi _ {s} \ left ({\ frac {z} {2}} \ right) + \ psi _ {s} \ left ({\ frac {z + 1} {2}} \ right) = 2 ^ {{s + 1}} \ psi _ {s} (z) + {\ frac {2 ^ {{s + 1}} \ ln 2} {\ Gamma (-s)}} \ zeta (s + 1, z)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dddd674e3a95438cf9b6364bdab4d1fca10cd89)
derive. A generalization of this is
![\ sum \ limits _ {{k = 0}} ^ {{n-1}} \ psi _ {s} \ left ({\ frac {z + k} {n}} \ right) = n ^ {{s +1}} \ psi _ {s} (z) + {\ frac {n ^ {{s + 1}} \ ln n} {\ Gamma (-s)}} \ zeta (s + 1, z),](https://wikimedia.org/api/rest_v1/media/math/render/svg/017cdeb2abb1d05dff6261640faadfd331ab1e68)
which is an equivalent to the Gaussian multiplication formula of the gamma function and contains the
multiplication formula as a special case for .
![{\ displaystyle s \ in \ mathbb {N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fde27eac1945df21b2ef2f5a15859e4e9c288698)
q-polygamma function
The polygamma function is defined by
![q](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)
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literature
credentials
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↑ Oliver Espinosa and Victor H. Moll:
A Generalized Polygamma Function on arXiv.org e-Print archive 2003.
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↑ Eric W. Weisstein : q-Polygamma Function . In: MathWorld (English).