QBinomial

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`QBinomial`

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QBinomial[n,m,q]

gives the -binomial coefficient .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
.
QBinomial automatically threads over lists.

Examples

open allclose all

Basic Examples (6)Summary of the most common use cases

Exact evaluation with numbers:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-bjr0rh

Out[1]=1

Plot over a subset of the reals:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-8vy2o

Out[1]=1

Plot over a subset of the complexes:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-kiedlx

Out[1]=1

Series expansion at the origin:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-fdkkja

Out[1]=1

Series expansion at Infinity:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-20imb

Out[1]=1

Use FunctionExpand to obtain Gaussian polynomials:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-puz0a3

Out[1]=1

Scope (20)Survey of the scope of standard use cases

Numerical Evaluation (5)

Evaluate numerically:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-l274ju

Out[1]=1

In[3]:=3

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https://wolfram.com/xid/0rtf7pybe-wlv0g

Out[3]=3

Evaluate to high precision:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-b0wt9

Out[1]=1

The precision of the output tracks the precision of the input:

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-y7k4a

Out[2]=2

Complex number inputs:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-hfml09

Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-di5gcr

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-bq2c6r

Out[2]=2

Compute the elementwise values of an array:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-thgd2

Out[1]=1

Or compute the matrix QBinomial function using MatrixFunction:

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-o5jpo

Out[2]=2

Specific Values (5)

QBinomial for symbolic parameters:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-fc9m8o

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-ojm7m3

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0rtf7pybe-di9w9p

Out[3]=3

Value at zero:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-e41pf2

Out[1]=1

Find the minimum of QBinomial[3,2,q]:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-f2hrld

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-dinc59

Out[2]=2

QBinomial threads elementwise over lists:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-d89eov

Out[1]=1

TraditionalForm formatting:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-bn5qfk

Visualization (2)

Plot the QBinomial function for various parameters:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-c0x9p4

Out[1]=1

Plot the real part of :

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-dbvuei

Out[1]=1

Plot the imaginary part of :

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-f2g90r

Out[2]=2

Function Properties (4)

has both singularities and discontinuities for and for :

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-mdtl3h

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-mn5jws

Out[2]=2

is neither non-negative nor non-positive:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-84dui

Out[1]=1

QBinomial is neither convex nor concave:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-nclwh8

Out[1]=1

TraditionalForm formatting:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-lzg9qr

Differentiation (2)

The first derivative with respect to n when m=1/2:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-krpoah

Out[1]=1

The first derivative with respect to m when n=1 and q=2:

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-frvu96

Out[2]=2

Higher derivatives with respect to m:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-z33jv

Out[1]=1

Plot the higher derivatives with respect to m when n=3 and q=2:

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-fxwmfc

Out[2]=2

Series Expansions (2)

Find the Taylor expansion using Series:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-ewr1h8

Out[1]=1

Plots of the first three approximations around :

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-binhar

Out[2]=2

The Taylor expansion at a generic point:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-jwxla7

Out[1]=1

Generalizations & Extensions (1)Generalized and extended use cases

QBinomial can be applied to a power series:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-cyyqa6

Out[1]=1

Applications (4)Sample problems that can be solved with this function

Explicit combinatorial construction of QBinomial:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-d6uz5g

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-mid1h4

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0rtf7pybe-lipq7c

Out[3]=3

-binomial is a generating function for the sequence in a grid-shading problem:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-kpo2xv

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-bb2br1

Out[2]=2

Compare to explicit counting:

In[3]:=3

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https://wolfram.com/xid/0rtf7pybe-clr28t

Out[3]=3

Elements in the -Pascal triangle satisfy two recurrence relations:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-b6lwhk

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-d01t63

Out[2]=2

The number of subspaces in the -dimensional vector space over with prime-power :

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-dp25so

Total number of subspaces in three-dimensional vector space over :

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-ghmbj

Out[2]=2

Check using recurrence equation for Galois numbers:

In[3]:=3

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https://wolfram.com/xid/0rtf7pybe-g1fvj2

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0rtf7pybe-bhlv7y

Out[4]=4

Properties & Relations (2)Properties of the function, and connections to other functions

Use FunctionExpand and FullSimplify to manipulate expressions containing QBinomial:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-cxrac1

Out[1]=1

Build series expansions:

In[1]:=1

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https://wolfram.com/xid/0rtf7pybe-jiirg

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0rtf7pybe-e8eyv7

Out[2]=2

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