Some definitions and formulae are represented[1].
Let be j, m, n natural numbers, k an integer number with |k| greater or equal n and p contained in the interval [0,1] (as probability for a step e.g. to the right). We define
The numbers Q0 (n, k) of the Q0 triangle correspond to the special case of same probabilities for steps to the right or to the left, i.e. for p=(1-p)=0.5:
The probabilities Q0Z (n) for return ("central meeting probabilities") correspond to the special case k = 0:
The Q1 triangle results from a superposition of two Q0 triangles with opposite sign, starting in position n=1, k=±1 after multiplication by 1/2. Addition of both means a "discrete differentiation" along k.
The absolute values |Q1 (n, k)| also arise, if after starting in row n=1 the following rows are constructed in usual way, but the numbers in the row centers k=0 are set to 0 in every row with even row number respectively, are so to speak "flown out", so that they can't be sources subsequently. Let for every even row number n be -Q2Z(n) "flowing out probability" there, i.e. the probability for flowing out centrally. Q2Z(n) is equivalent to the 1nd (discrete) derivative of Q1(n,k) in k=0 along k, i.e. Q2Z(n) = (Q1(n-1,1)-Q1(n-1,-1))/2; so Q2Z(n) is in k=0 the 2nd derivative of Q0(n,k) along k. It holds:
Q0M(n,k) is in case of odd n antisymmetric and
in case of even n symmetric regarding to k=0 .
So addition behavior of right and left side is like the one of amplitudes of fermions resp. bosons.
Similarly to the analytical case multiple discrete differentiation can be defined recursively (by formation of higher-order finite differences). Let be QDP(d,n,k) the d times along k differentiated function Q0P (n, k, p), then
and for n ³ d ³ 1
.
At this n ³ d is necessary that enough values are available to build a (finite) difference of d-th order.
(BinCoeffDiffMatrix)The representation of operators as matrices is often useful in discrete considerations. Here a matrix representation of the operator for discrete differentiation in form of an example matrix with high number of dimensions for clarification of the development: Multiplication of a 15-dimensional vector by the following matrix
¦ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 ¦
¦ -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ¦
¦ 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 ¦
¦ 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 ¦
¦ 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 ¦
¦ 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 ¦
¦ 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 ¦
D := ¦ 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 ¦ * 1/2
¦ 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 ¦
¦ 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 ¦
¦ 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 ¦
¦ 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 ¦
¦ 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 ¦
¦ 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 ¦
¦ 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 ¦
means first order discrete differentiation "along" the index k of the vector components (calculation of the finite first order difference - the shift dk of the index k is 2, therefore division by 2). Multiplication by the n-ten power D^n of this matrix yields n-th order discrete differentiation (formation of the finite n-th order difference). For example means multiplication by
¦ -20 0 15 0 -6 0 1 0 0 1 0 -6 0 15 0 ¦
¦ 0 -20 0 15 0 -6 0 1 0 0 1 0 -6 0 15 ¦
¦ 15 0 -20 0 15 0 -6 0 1 0 0 1 0 -6 0 ¦
¦ 0 15 0 -20 0 15 0 -6 0 1 0 0 1 0 -6 ¦
¦ -6 0 15 0 -20 0 15 0 -6 0 1 0 0 1 0 ¦
¦ 0 -6 0 15 0 -20 0 15 0 -6 0 1 0 0 1 ¦
6 ¦ 1 0 -6 0 15 0 -20 0 15 0 -6 0 1 0 0 ¦
D = ¦ 0 1 0 -6 0 15 0 -20 0 15 0 -6 0 1 0 ¦ * 1/64
¦ 0 0 1 0 -6 0 15 0 -20 0 15 0 -6 0 1 ¦
¦ 1 0 0 1 0 -6 0 15 0 -20 0 15 0 -6 0 ¦
¦ 0 1 0 0 1 0 -6 0 15 0 -20 0 15 0 -6 ¦
¦ -6 0 1 0 0 1 0 -6 0 15 0 -20 0 15 0 ¦
¦ 0 -6 0 1 0 0 1 0 -6 0 15 0 -20 0 15 ¦
¦ 15 0 -6 0 1 0 0 1 0 -6 0 15 0 -20 0 ¦
¦ 0 15 0 -6 0 1 0 0 1 0 -6 0 15 0 -20 ¦
6-th order discrete Differentiation resp. calculation of the finite 6-th order difference. The rows resp. columns of the matrix D^n contain the binomial coefficients, divided by 2^n, in this example the numbers 6!/(k!·(6 - k)!·2^6) = Q0(6, 2k-6).
horizontal (along localization):
vertical (along time):
Correspondence in the middle:
.
Let be d,l ³ n.
We define the weighted scalar product QSP by
.
Then for d¹l is valid
(i.e. orthogonality) and otherwise
,
particularly
.
The denominator in the last expression corresponds to the number of the way possibilities from point (0,0) to point (n,n-2d) in the Q0 triangle.
This chapter contains some elementary formulae for variably p (and n>0).
The first finite difference (discrete derivative) of Q0P is
In the border the probabilities p and 1-p are very different. With p->0 also v->0 (low temperature).
In the center the probabilities p and 1-p are nearly equal, i.e. p-> 1/2 and with that v-> C (the borderline case p=1/2 resp. v=C is represented by Q0 and Q1).
Let be
i.e.
then is valid for every sequence (kn) with (kn)^3/n^2®0 für n®¥ (p. 80 [likr])
(DiracDeltaFu)It is
The behavior for n®¥ can be illustrated by a n proportional scale fitting, i.e. by a horizontal compression and a vertical stretching by respectively the factor n. This doesn't touch the value of the integral:
For n®¥ therefore the function f(x) = n Q0E(n, nx) / 2 converges towards the Dirac delta-function.
(HermPol)The Hermite polynomials Hn(x) are (except sign) special cases of pre-factors resulting from multiple differentiation: