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Fungrim entry: 926b36

∣R−a ⁣([β,…,β⏟n times],[z1,z2,…,zn])−A−a∑N=0K−1(a)N(nβ)NTN ⁣([β,…,β⏟n times],[z1,z2,…,zn])∣≤∣A−a∣(∣a∣)KMKK!(1−M)max⁡(∣a∣,1)   where A=1n∑k=1nzk,  Zk=1−zkA,  M=max⁡ ⁣(∣Z1∣,∣Z2∣,…,∣Zn∣)\left|R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) - {A}^{-a} \sum_{N=0}^{K - 1} \frac{\left(a\right)_{N}}{\left(n \beta\right)_{N}} T_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)\right| \le \frac{\left|{A}^{-a}\right| \left(\left|a\right|\right)_{K} {M}^{K}}{K ! {\left(1 - M\right)}^{\max\left(\left|a\right|, 1\right)}}\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k},\;Z_{k} = 1 - \frac{z_{k}}{A},\;M = \max\!\left(\left|Z_{1}\right|, \left|Z_{2}\right|, \ldots, \left|Z_{n}\right|\right)
Assumptions:a∈R  and⁡  β∈(0,∞)  and⁡  n∈Z≥1  and⁡  K∈Z≥1  and⁡  (zk∈C  and⁡  ∣1−nzk∑j=1nzj∣<1   for all k∈{1,2,…,n})  and⁡  (Re⁡ ⁣(zk)>0   for all k∈{1,2,…,n})a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \beta \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; K \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|1 - \frac{n z_{k}}{\sum_{j=1}^{n} z_{j}}\right| < 1 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \left(\operatorname{Re}\!\left(z_{k}\right) > 0 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right)
References:
  • https://doi.org/10.6028/jres.107.034
TeX:
\left|R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) - {A}^{-a} \sum_{N=0}^{K - 1} \frac{\left(a\right)_{N}}{\left(n \beta\right)_{N}} T_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)\right| \le \frac{\left|{A}^{-a}\right| \left(\left|a\right|\right)_{K} {M}^{K}}{K ! {\left(1 - M\right)}^{\max\left(\left|a\right|, 1\right)}}\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k},\;Z_{k} = 1 - \frac{z_{k}}{A},\;M = \max\!\left(\left|Z_{1}\right|, \left|Z_{2}\right|, \ldots, \left|Z_{n}\right|\right)

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \beta \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; K \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \left(z_{k} \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|1 - \frac{n z_{k}}{\sum_{j=1}^{n} z_{j}}\right| < 1 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right) \;\mathbin{\operatorname{and}}\; \left(\operatorname{Re}\!\left(z_{k}\right) > 0 \;\text{ for all } k \in \{1, 2, \ldots, n\}\right)
Definitions:
Fungrim symbol Notation Short description
Abs∣z∣\left|z\right| Absolute value
CarlsonHypergeometricRR−a ⁣(b,z)R_{-a}\!\left(b, z\right) Carlson multivariate hypergeometric function
Powab{a}^{b} Power
Sum∑nf(n)\sum_{n} f(n) Sum
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
CarlsonHypergeometricTTN ⁣(b,z)T_{N}\!\left(b, z\right) Term in expansion of Carlson multivariate hypergeometric function
Factorialn!n ! Factorial
RRR\mathbb{R} Real numbers
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity∞\infty Positive infinity
ZZGreaterEqualZ≥n\mathbb{Z}_{\ge n} Integers greater than or equal to n
CCC\mathbb{C} Complex numbers
Range{a,a+1,…,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
ReRe⁡(z)\operatorname{Re}(z) Real part
Source code for this entry:
Entry(ID("926b36"),
    Formula(Where(LessEqual(Abs(Sub(CarlsonHypergeometricR(Neg(a), List(Repeat(beta, n)), List(z_(k), For(k, 1, n))), Mul(Pow(A, Neg(a)), Sum(Mul(Div(RisingFactorial(a, N), RisingFactorial(Mul(n, beta), N)), CarlsonHypergeometricT(N, List(Repeat(beta, n)), List(z_(k), For(k, 1, n)))), For(N, 0, Sub(K, 1)))))), Div(Mul(Mul(Abs(Pow(A, Neg(a))), RisingFactorial(Abs(a), K)), Pow(M, K)), Mul(Factorial(K), Pow(Sub(1, M), Max(Abs(a), 1))))), Def(A, Mul(Div(1, n), Sum(z_(k), For(k, 1, n)))), Def(Z_(k), Sub(1, Div(z_(k), A))), Def(M, Max(Step(Abs(Z_(k)), For(k, 1, n)))))),
    Variables(a, beta, z_, n, K),
    Assumptions(And(Element(a, RR), Element(beta, OpenInterval(0, Infinity)), Element(n, ZZGreaterEqual(1)), Element(K, ZZGreaterEqual(1)), All(And(Element(z_(k), CC), Less(Abs(Sub(1, Div(Mul(n, z_(k)), Sum(z_(j), For(j, 1, n))))), 1)), ForElement(k, Range(1, n))), All(Greater(Re(z_(k)), 0), ForElement(k, Range(1, n))))),
    References("https://doi.org/10.6028/jres.107.034"))

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2021-03-15 19:12:00.328586 UTC