({{, (RefLink[AugmentedSymmetricPolynomial, paclet:ref/AugmentedSymmetricPolynomial])[{(Subscript[r, 1]),(Subscript[r, 2]),…}] 
represents a formal augmented symmetric polynomial with exponents (Subscript[r, 1]), (Subscript[r, 2]), ….}, {, (RefLink[AugmentedSymmetricPolynomial, paclet:ref/AugmentedSymmetricPolynomial])[{{(Subscript[r, 11]),…,(Subscript[r, 1n])},{(Subscript[r, 21]),…,(Subscript[r, 2n])},…}]
represents a multivariate formal augmented symmetric polynomial with exponent vectors {(Subscript[r, 11]), …, (Subscript[r, 1n])}, {(Subscript[r, 21]), …, (Subscript[r, 2n])}, ….}, {, (RefLink[AugmentedSymmetricPolynomial, paclet:ref/AugmentedSymmetricPolynomial])[rspec,data]
gives the augmented symmetric polynomial in data.}})

OpenersNotesSection(Spacer1)Details

(RefLink[AugmentedSymmetricPolynomial, paclet:ref/AugmentedSymmetricPolynomial])[{(Subscript[r, 1]),…,(Subscript[r, k])},{(Subscript[x, 1]),…,(Subscript[x, n])}] is given by ∑(Subscript[x, (Subscript[i, 1])]^(Subscript[r, 1]))⋯ (Subscript[x, (Subscript[i, k])]^(Subscript[r, k])), where the sum is taken over the index set (Subscript[i, 1])≠(Subscript[i, 2])≠⋯≠(Subscript[i, k]) and 0≤(Subscript[i, 1]),…,(Subscript[i, k])≤n.

In the multivariate case, (Subscript[r, i]) and (Subscript[x, i]) are vectors and ((Subscript[x, i])^(Subscript[r, i])) is interpreted as ((Subscript[x, i1])^(Subscript[r, i1]))((Subscript[x, i2])^(Subscript[r, i2]))⋯.

(RefLink[AugmentedSymmetricPolynomial, paclet:ref/AugmentedSymmetricPolynomial])[rspec] can be used to represent formal augmented symmetric sums used in moment estimators.

(RefLink[MomentConvert, paclet:ref/MomentConvert]) can be used to generate moment estimators in terms of (RefLink[AugmentedSymmetricPolynomial, paclet:ref/AugmentedSymmetricPolynomial]) objects.

(RefLink[MomentEvaluate, paclet:ref/MomentEvaluate]) can be used to evaluate polynomials of formal (RefLink[AugmentedSymmetricPolynomial, paclet:ref/AugmentedSymmetricPolynomial]) objects on a dataset.

OpenersPrimaryExamplesSection(Spacer1)Examples  (5)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

AugmentedSymmetricPolynomial[{1,2},{x,y,z}]

AugmentedSymmetricPolynomial[{{1,2},{3,4}},{{(Subscript[x, 1]),(Subscript[x, 2])},{(Subscript[y, 1]),(Subscript[y, 2])},{(Subscript[z, 1]),(Subscript[z, 2])}}]

OpenersExampleSection1(Spacer1)Scope  (2)

Use (RefLink[MomentEvaluate, paclet:ref/MomentEvaluate]) to evaluate formal augmented symmetric polynomials on data:

3AugmentedSymmetricPolynomial[{1,2}]-AugmentedSymmetricPolynomial[{1}]^3

MomentEvaluate[%,{(Subscript[x, 1]),(Subscript[x, 2]),(Subscript[x, 3])}]

(RefLink[TraditionalForm, paclet:ref/TraditionalForm]) formatting:

AugmentedSymmetricPolynomial[{r1,r2,r3}]//TraditionalForm

AugmentedSymmetricPolynomial[{{p1,q1},{p2,q2},{p3,q3}}]//TraditionalForm

OpenersExampleSection3(Spacer1)Applications  (1)

Linearize polynomials in (RefLink[AugmentedSymmetricPolynomial, paclet:ref/AugmentedSymmetricPolynomial]):

MomentConvert[AugmentedSymmetricPolynomial[{1,1}]AugmentedSymmetricPolynomial[{2,2}],PowerSymmetricPolynomial,AugmentedSymmetricPolynomial]

OpenersExampleSection4(Spacer1)Properties & Relations  (1)

(RefLink[AugmentedSymmetricPolynomial, paclet:ref/AugmentedSymmetricPolynomial]) with a single exponent is equivalent to (RefLink[PowerSymmetricPolynomial, paclet:ref/PowerSymmetricPolynomial]):

AugmentedSymmetricPolynomial[{r}]

MomentConvert[%,PowerSymmetricPolynomial]

This relationship also holds for the multivariate generalization:

AugmentedSymmetricPolynomial[{{p,q}}]//MomentConvert[#,PowerSymmetricPolynomial]&

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[MomentConvert, paclet:ref/MomentConvert])  ▪  (RefLink[MomentEvaluate, paclet:ref/MomentEvaluate])  ▪  (RefLink[PowerSymmetricPolynomial, paclet:ref/PowerSymmetricPolynomial])  ▪  (RefLink[SymmetricPolynomial, paclet:ref/SymmetricPolynomial])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Statistical Moments and Generating Functions, paclet:guide/StatisticalMomentsAndGeneratingFunctions])}})}})