In the mathematical branch of combinatorics , the Bell polynomials , named after Eric Temple Bell , denote the following triangular arrangement of polynomials
-
,
where the sum is formed over all sequences of non-negative integers such that
![{\ displaystyle j_ {1}, j_ {2}, \ dots, j_ {n-k + 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd06633eed0231d211fedcf0bc0fef05f7f5d1d)
![j_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c4b258245d4ce139e3a77a525eb7efd4a6e270)
-
and .![{\ displaystyle 1 \, j_ {1} \; + \; 2 \, j_ {2} \; + \; 3 \, j_ {3} \; + \ cdots = n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48ef4fd23dc69e6d9985931f75a5a841d0c00112)
The Bell polynomial is a polynomial in variables . Its coefficients are whole numbers .
![B _ {{n, k}} (x_ {1}, x_ {2}, \ dots, x _ {{n-k + 1}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dbd1af656b8288b6d61e72f68be99c18cc96282)
![{\ displaystyle x_ {1}, x_ {2}, \ dots, x_ {n-k + 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb56314566a8607c119b087c8399fde48f925ac0)
Complete Bell polynomials
The sum
![B_ {n} (x_ {1}, \ dots, x_ {n}) = \ sum _ {{k = 1}} ^ {n} B _ {{n, k}} (x_ {1}, x_ {2 }, \ dots, x _ {{n-k + 1}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/76991d0ee74bbcc3c1dd06b481c3136c873d28a4)
is sometimes called the tes complete Bell polynomial . For better differentiation from the complete Bell polynomials, the polynomials defined above are sometimes referred to as incomplete or partial Bell polynomials.
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![B _ {{n, k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b70a55a02dca54fe8db8174fdf040477feff8d0f)
The complete Bell polynomials satisfy the following equation
![{\ displaystyle B_ {n} (x_ {1}, \ dots, x_ {n}) = \ det {\ begin {bmatrix} x_ {1} & {\ binom {n-1} {1}} x_ {2 } & {\ binom {n-1} {2}} x_ {3} & {\ binom {n-1} {3}} x_ {4} & {\ binom {n-1} {4}} x_ { 5} & \ cdots & \; \; x_ {n} \\\\ - 1 & x_ {1} & {\ binom {n-2} {1}} x_ {2} & {\ binom {n-2} { 2}} x_ {3} & {\ binom {n-2} {3}} x_ {4} & \ cdots & \; \; x_ {n-1} \\\\ 0 & -1 & x_ {1} & { \ binom {n-3} {1}} x_ {2} & {\ binom {n-3} {2}} x_ {3} & \ cdots & \; \; x_ {n-2} \\\\ 0 & 0 & -1 & x_ {1} & {\ binom {n-4} {1}} x_ {2} & \ cdots & \; \; x_ {n-3} \\\\ 0 & 0 & 0 & -1 & x_ {1} & \ cdots & \; \; x_ {n-4} \\\\\ vdots & \ vdots & \ vdots & \ vdots & \ ddots & \ ddots & \; \; \ vdots \\\\ 0 & 0 & 0 & 0 & \ cdots & -1 & \ ; \; x_ {1} \ end {bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/976f14f6db729ce3967fd4b6dcf1200dc17df82d)
Recursive representations
A recursive definition of the Bell polynomials is:
![{\ displaystyle B_ {0,0} ()}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac76d0e2df4e4de61ada1bcff8959fc5af146a6) |
,
|
![{\ displaystyle B_ {n, 0} (x_ {1}, \ dots, x_ {n + 1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f009a4c4c1f949d72471b362261f3a3e1a627853) |
![{\ displaystyle = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc9e66de468806365c20e32e83456cc526ce29e) |
For |
,
|
![{\ displaystyle B_ {n, k} (x_ {1}, \ dots, x_ {n-k + 1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8d94b3715a13f3044daaf19b149e8bd4b77e3b) |
![{\ displaystyle = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc9e66de468806365c20e32e83456cc526ce29e) |
For |
|
and
![{\ displaystyle B_ {n, k} (x_ {1}, \ dots, x_ {n-k + 1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8d94b3715a13f3044daaf19b149e8bd4b77e3b) |
![{\ displaystyle = \ sum _ {i = 1} ^ {n-k + 1} {\ binom {n-1} {i-1}} B_ {ni, k-1} (x_ {1}, \ dots , x_ {ni-k + 2}) \, x_ {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd34bb610e66e63de451d34b977f4043825daa05) |
For |
.
|
The complete Bell polynomials can be defined recursively as follows
|
and
![{\ displaystyle B_ {n + 1} (x_ {1}, \ ldots, x_ {n + 1}) = \ sum _ {i = 0} ^ {n} {\ binom {n} {i}} B_ { ni} (x_ {1}, \ ldots, x_ {ni}) \, x_ {i + 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75873bab6dec5b4b8542b47d25511c3a5c67c693) |
for .
![{\ displaystyle n \ geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0) |
They also satisfy the following recursive differential formula:
![{\ displaystyle {\ begin {aligned} B_ {n} (x_ {1}, \ ldots, x_ {n}) = {\ frac {1} {n-1}} \ left [\ sum _ {i = 2 } ^ {n} \ right. & \ sum _ {j = 1} ^ {i-1} (i-1) {\ binom {i-2} {j-1}} x_ {j} x_ {ij} {\ frac {\ partial B_ {n-1} (x_ {1}, \ dots, x_ {n-1})} {\ partial x_ {i-1}}} \\ [5pt] & \ left. { } + \ sum _ {i = 2} ^ {n} \ sum _ {j = 1} ^ {i-1} {\ frac {x_ {i + 1}} {\ binom {i} {j}}} {\ frac {\ partial ^ {2} B_ {n-1} (x_ {1}, \ dots, x_ {n-1})} {\ partial x_ {j} \ partial x_ {ij}}} \ right . \\ [5pt] & \ left. {} + \ Sum _ {i = 2} ^ {n} x_ {i} {\ frac {\ partial B_ {n-1} (x_ {1}, \ dots, x_ {n-1})} {\ partial x_ {i-1}}} \ right]. \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8063f0be30dd504b0ae58533a3ce24749fcc34)
Combinatorial meaning
A non-negative integer is given as the number of elements of the set to be partitioned.
![{\ displaystyle n \ in \ mathbb {N} _ {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf5c8ad993619a325ca57a25c22cdc75a460f88)
If the whole number (= a quantity of the size) is broken down into a sum of summands (= partitions) , in which the summand (= the partition size) occurs once, twice and the summand times, then the number corresponds to the possible partition ations that can be formed with a -elementary set, the coefficient of the monomial in the Bell polynomial that can be assigned to the partition sizes . is then the polynomial of all monomials with the total degree and the quantity .![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![{\ displaystyle j_ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a671019a672cd149a8d1584861e2f8fe9728a9e)
![{\ displaystyle j_ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f29268662641c1e61d6faba764a493c74ba962b)
![j_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c4b258245d4ce139e3a77a525eb7efd4a6e270)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle (1,2, \ dots, k)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1482115d6f453a4ff9a1e2cc9b36c499dbeb5333)
![{\ displaystyle x_ {1} ^ {j_ {1}} \ cdots x_ {k} ^ {j_ {k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74dbf04d4223fdf1e2b714451757d19f7c1839ae)
![{\ displaystyle k = j_ {1} + j_ {2} + j_ {3} + \ cdots = k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49be7ce1db3834e14b4ce9a1fc581ed104c80a43)
The names (actually: the numbers) of the indefinite |
![x_ {1},](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb7118b371b245f6611bd7a6c8ee319dbdc32825) |
![x_ {2},](https://wikimedia.org/api/rest_v1/media/math/render/svg/e66e41f50371ed4b1910b485f2e2d0eb28f11533) |
![{\ displaystyle \ dots,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4dec7c45e485b15b103a2b0974678a3d49b9032) |
|
act only as a post for pinning the number |
![{\ displaystyle j_ {1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/443605ef08d68ca0c6d00b1fafb715bd111037a1) |
![{\ displaystyle j_ {2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00d8047b1d9a39aa0faa315f224a6958ac045041) |
![{\ displaystyle \ dots,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4dec7c45e485b15b103a2b0974678a3d49b9032) |
|
of partitions in the partitioning that exactly |
summand
|
![1,](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5fd8163a83100c5330622e9e317fa4e872403) |
![{\ displaystyle 2,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/993d53082bc05c266933af9a892e1ce2128547cd) |
![{\ displaystyle \ dots,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4dec7c45e485b15b103a2b0974678a3d49b9032) |
![{\ displaystyle n \! - \! k \! + \! 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f500ce8de152d4233444269ce20076147ebb6d67) |
![\}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cf208e5d370391e66767f13641bd5ee6ad93825) |
Elements should have
|
as an exponent of the power .
![{\ displaystyle x_ {i} ^ {j_ {i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a1c4b92f24a93ad87aafc30467d77bdb60960f5) |
An exponent of 1 is usually not noted. If the exponent is 0, then the whole power is suppressed. The largest partition size in partitions is which partition size occurs exactly . The smallest partition size (= 1) then occurs exactly once in this partitioning .
![x_i ^ 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/6319e9660975eba91bf542b34f0d68748f6c450e)
![k](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
![{\ displaystyle n \! - \! k \! + \! 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f500ce8de152d4233444269ce20076147ebb6d67)
![{\ displaystyle j_ {n-k + 1} = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23112e06e093f138692dd43578d9d711ee8eac16)
![{\ displaystyle j_ {1} = k-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e0ecb6c0abe68a1a57416a689c6ce21aab33d14)
The preferred order of the monomials in the Bell polynomial is the lexicographically ascending order with the lowest rank for , so occurs occurs .
![x_ {1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308)
![{\ displaystyle x_ {1} ^ {\, 2} x_ {2} = x_ {1} x_ {1} x_ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1380501c4936acd111f353630b11b2d11bddb45)
![{\ displaystyle x_ {1} x_ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b72aaacad232ae448753193b1f80b434bc61b88)
![x_ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766)
- Examples
-
for .![n \ geq 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe)
-
for .![k> n](https://wikimedia.org/api/rest_v1/media/math/render/svg/66e81682bf174c978e9008ffb557ba4da2cf7478)
-
for .![{\ displaystyle n \ geq 1, k \ leq n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/814761ca5704d6d8357b53ccc94714715cee7ee0)
- Furthermore is
-
,
- because it
- (Monom ) There are 6 ways to partition a set of elements into partitions with 1 and 5 elements,
![{\ displaystyle 6 \, x_ {1} x_ {5} \ longrightarrow}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33d38bfde5c8ab9e56e9d15316db537abd3a8ac0)
![{\ displaystyle n = 6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f)
![{\ displaystyle k = 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd301789e1f25a3da4be297ff637754ebee5f5d)
- (Monom ) 15 ways are to partition a lot of elements into partitions with 2 and 4 elements, and it
![{\ displaystyle 15 \, x_ {2} x_ {4} \ longrightarrow}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8597f120f30117b0b25a34a3ec66d222210faa58)
![{\ displaystyle n = 6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f)
![{\ displaystyle k = 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd301789e1f25a3da4be297ff637754ebee5f5d)
- (Monom ) 10 ways to partition a lot of elements into partitions with 3 and 3 elements.
![{\ displaystyle 10 \, x_ {3} ^ {2} \ longrightarrow}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9580030307adba9a91f3a0d5d42e6dfd3c09997)
![{\ displaystyle n = 6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f)
![{\ displaystyle k = 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd301789e1f25a3da4be297ff637754ebee5f5d)
![{\ displaystyle B_ {6,3} (x_ {1}, x_ {2}, x_ {3}, x_ {4}) = 15x_ {1} ^ {2} x_ {4} + 60x_ {1} x_ { 2} x_ {3} + 15x_ {2} ^ {3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc2a681231e93d35818ff9c38aa57a0d9929d6cb)
- because it
- (Monom ) there are 15 ways to partition a set of elements into partitions with 1, 1 and 4 elements each,
![{\ displaystyle 15 \, x_ {1} ^ {2} x_ {4} \ longrightarrow}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a938084a0f689929269ba3fabb1e7787bdc267f5)
![{\ displaystyle n = 6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f)
![{\ displaystyle k = 3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/662e06a2436f8a44fec791f5c794621f10dc8f30)
- (Monom ) 60 ways to partition a lot of elements into partitions of 1, 2 and 3 elements each, and there
![{\ displaystyle 60 \, x_ {1} x_ {2} x_ {3} \ longrightarrow}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc94b86daa6a3c74cbfabfce6491491e7071635e)
![{\ displaystyle n = 6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f)
![{\ displaystyle k = 3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/662e06a2436f8a44fec791f5c794621f10dc8f30)
- (Monom ) 15 possibilities exist to partition a set of elements into partitions with 2, 2 and 2 elements each.
![{\ displaystyle 15 \, x_ {2} ^ {3} \ longrightarrow}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c98da049baebb1b148007de196c8a39fb37209c2)
![{\ displaystyle n = 6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f)
![{\ displaystyle k = 3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/662e06a2436f8a44fec791f5c794621f10dc8f30)
properties
![B _ {{n, k}} (1!, 2!, \ Dots, (n-k + 1)!) = {\ Binom {n} {k}} {\ binom {n-1} {k-1 }} (nk)!](https://wikimedia.org/api/rest_v1/media/math/render/svg/909c02d638e6bfdbf1876a385daebddd5b35fb33)
Bell polynomials and Stirling numbers
The value of the Bell polynomial when all are equal to 1 is a Stirling number of the second kind![{\ displaystyle B_ {n, k} (x_ {1}, x_ {2}, \ dots)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91fe91d704b2db2c50f0723e4bb4f6cc4ea326d2)
-
.
The sum
![\ sum _ {{k = 1}} ^ {n} B _ {{n, k}} (1,1,1, \ dots) = \ sum _ {{k = 1}} ^ {n} \ left \ {{n \ atop k} \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f2c446a6973dee10f996afb47d353d45944751c)
corresponding to the th Bell number representing the number of possible partitions of a quantity with describes elements.
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
Folding identity
A kind of convolution can be defined for consequences and :
![{\ displaystyle (x_ {n}) _ {n = 1,2, \ dotsc}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfda2c86c4c40a114e961ba753ae32d76a9c073)
![{\ displaystyle (y_ {n}) _ {n = 1,2, \ dotsc}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb6195f91f68b42654c93ed84c53973f7d4db2e)
-
,
where the limits are the sum and instead of and .
![1](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf)
![n-1](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521)
![{\ displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
Be the th term of the sequence
![{\ displaystyle x_ {n} ^ {k \ diamond}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/084858a783381acbc7e1ef6c7887201fe61041ac)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
-
,
then applies:
-
.
The convolutional identity can be used to compute single Bell polynomials. The calculation of results from
![B _ {{4,3}} (x_ {1}, x_ {2})](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca6620f437cdfc1fae7ecd5fd21e9f76cd95d5cf)
![x = (x_ {1} \, \ x_ {2} \, \ x_ {3} \, \ x_ {4} \, \ dots)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7835992b71cb6146d765266b1a535dab038cd59)
![{\ displaystyle x \ diamond x = (0, \ 2x_ {1} ^ {2} \, \ 6x_ {1} x_ {2} \, \ 8x_ {1} x_ {3} + 6x_ {2} ^ {2 } \, \ dots)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63db3a55536723b11e581c93b86cf760e6cb6beb)
![{\ displaystyle x \ diamond x \ diamond x = (0 \, \ 0 \, \ 6x_ {1} ^ {3} \, \ 36x_ {1} ^ {2} x_ {2} \, \ dots)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa0a9d56da575378d36fb5b43f8168e54f23a85)
and accordingly,
![{\ displaystyle B_ {4,3} (x_ {1}, x_ {2}) = {\ frac {(x \ diamond x \ diamond x) _ {4}} {3!}} = 6x_ {1} ^ {2} x_ {2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3332d84e47f437f800ceba1a7c0068200bfd3c73)
Applications
Formula by Faà di Bruno
The Faà di Bruno's formula can use the Bell polynomials as follows expressed are:
![{d ^ {n} \ over dx ^ {n}} f (g (x)) = \ sum _ {{k = 0}} ^ {n} f ^ {{(k)}} (g (x) ) B _ {{n, k}} \ left (g '(x), g' '(x), \ dots, g ^ {{(n-k + 1)}} (x) \ right).](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea83c08f3a6a28b816eca476ed8318a3d27e6627)
A power series version of Faà di Bruno's formula can be set up in a similar way . Accepted
![f (x) = \ sum _ {{n = 1}} ^ {\ infty} {a_ {n} \ over n!} x ^ {n} \ qquad {\ mathrm {and}} \ qquad g (x) = \ sum _ {{n = 1}} ^ {\ infty} {b_ {n} \ over n!} x ^ {n}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/23f9024b76710e9ef60a9e2e524b9daa3b567c26)
Then
![g (f (x)) = \ sum _ {{n = 1}} ^ {\ infty} {\ sum _ {{k = 1}} ^ {{n}} b_ {k} B _ {{n, k }} (a_ {1}, \ dots, a _ {{n-k + 1}}) \ over n!} x ^ {n}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/513089423999e76c380395d1b7b547f129918c3f)
The complete Bell polynomials appear in the exponential function of a formal power series :
-
.
Moments and accumulators
The sum
![B_ {n} (\ kappa _ {1}, \ dots, \ kappa _ {n}) = \ sum _ {{k = 1}} ^ {n} B _ {{n, k}} (\ kappa _ { 1}, \ dots, \ kappa _ {{n-k + 1}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/7eff29a8a62a91d2b11b8b78a590f860c5887d5c)
is the th moment of a probability distribution whose first are cumulants . In other words, the th moment is the th complete Bell polynomial evaluated on the first cumulant.
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle \ kappa _ {1}, \ dots, \ kappa _ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7cfe4f9c18ec4c80f21ba0bdabb5caf13a799f2)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
Representation of polynomial sequences with binomial properties
For any (scalar) sequence: let
![{\ displaystyle a_ {1}, a_ {2}, a_ {3}, \ dots}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b7eb2acf428a2c57ea6c3a379744fcce1d99c0)
-
.
This polynomial sequence satisfies the binomial property, i. H.
![p_ {n} (x + y) = \ sum _ {{k = 0}} ^ {n} {n \ choose k} p_ {k} (x) p _ {{nk}} (y)](https://wikimedia.org/api/rest_v1/media/math/render/svg/13b5052c2fa895d69387ad1fffc41150ea18ab7a)
for . It is true that all polynomial sequences which satisfy the binomial property are of this form.
![n \ ge 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0)
For the inverse of the composition of
the formal power series
![h (x) = \ sum _ {{n = 1}} ^ {\ infty} {a_ {n} \ over n!} x ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad67991b2ae95178eacf60cc909d5f272b175162)
arises for everyone
![{\ displaystyle h ^ {- 1} {\ bigl (} p_ {n} ^ {\ prime} (x) {\ bigr)} = np_ {n-1} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5daa2dcc34448f5c0e52ccc0a2158a18bda8e821)
with the above polynomials with coefficients in .
![p_n (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a47cca302b556e703b68e2a12168117208f47ff)
![{\ displaystyle a_ {1}, \ dots, a_ {n-k + 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77b13f6cbeb0a318a62fd9ae71284dcd9454a941)
literature
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Eric Temple Bell : Partition Polynomials . In: Annals of Mathematics . tape 29 , no. 1/4 , 1927, pp. 38-46 , doi : 10.2307 / 1967979 , JSTOR : 1967979 .
- Louis Comtet: Advanced Combinatorics: The Art of Finite and Infinite Expansions . Reidel Publishing Company, Dordrecht, Holland / Boston, US 1974.
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Steven Roman : The Umbral Calculus . Academic Press, 1984, ISBN 0-08-087430-4 ( 123library.org ).
- Vassily G. Voinov, Mikhail S. Nikulin: On power series, Bell polynomials, Hardy-Ramanujan-Rademacher problem and its statistical applications . In: Cybernetics . tape 30 , no. 3 , 1994, ISSN 0023-5954 , pp. 343–358 ( kybernetika.cz [PDF]).
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George Andrews : The Theory of Partitions (= Cambridge Mathematical Library ). 1st edition. Cambridge University Press, 1998, ISBN 0-521-63766-X , pp. 204-211 .
- Silvia Noschese, Paolo E. Ricci: Differentiation of Multivariable Composite Functions and Bell Polynomials . In: Journal of Computational Analysis and Applications . tape 5 , no. 3 , 2003, p. 333-340 , doi : 10.1023 / A: 1023227705558 .
- Moncef Abbas, Sadek Bouroubi: On new identities for Bell's polynomial . In: Disc. Math . tape 293 , 2005, pp. 5-10 , doi : 10.1016 / j.disc.2004.08.023 .
- Khristo N. Boyadzhiev: Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals . In: Abstract and Applied Analysis . 2009, doi : 10.1155 / 2009/168672 .
- Martin Griffiths: Families of sequences from a class of multinomial sums . In: Journal of Integer Sequences . tape 15 , 2012 ( cs.uwaterloo.ca ).
Individual evidence
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↑ Nikita Alexeev, Anna Pologova, Max A. Alekseyev: Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs . In: Journal of Computational Biology . 24, No. 2, 2017, pp. 93-105. arxiv : 1503.05285 . doi : 10.1089 / cmb.2016.0190 . , sect. 4.2