Meromorphic Stirling numbers and Bessel numbers

DefinitionThe meromorphic Stirling numbers \(S_{2;-1}(n,k)\)#✓fully proved

For natural numbers \(n\) and \(k\) with \(1 \le k \le n\), the meromorphic Stirling number \(S_{2;-1}(n,k)\) is the rational number

\[S_{2;-1}(n,k) = \frac{(-1)^{n-k}\,(n-1)!}{2^{n-k}\,(k-1)!}\binom{2n-k-1}{n-1}.\]

For other \(n\) and \(k\) the same expression is evaluated with natural-number subtraction truncated at \(0\).

public def meromorphicStirlingFirst (n k : ℕ) : ℚ :=
  (((-1 : ℚ) ^ (n - k) * (((n - 1).factorial : ℕ) : ℚ)) /
    (((2 : ℚ) ^ (n - k)) * ((((k - 1).factorial : ℕ)) : ℚ))) *
    (((Nat.choose (2 * n - k - 1) (n - 1) : ℕ)) : ℚ)

Sources

• Schork, On Generalized Stirling Numbers and Normal Ordering, StirW1 formula. Defined independently from the Bessel side.

Used by 1

Used byFirst identification of meromorphic Stirling and Bessel numbers

DefinitionThe meromorphic Stirling numbers \(S_{-1;1}(n,k)\)#✓fully proved

For natural numbers \(n\) and \(k\) with \(k \le n \le 2k\), the meromorphic Stirling number \(S_{-1;1}(n,k)\) is the rational number

\[S_{-1;1}(n,k) = \frac{(2n-2k)!}{2^{n-k}\,(n-k)!}\binom{n}{2k-n}.\]

For other \(n\) and \(k\) the same expression is evaluated with natural-number subtraction truncated at \(0\).

public def meromorphicStirlingSecond (n k : ℕ) : ℚ :=
  ((((2 * n - 2 * k).factorial : ℕ)) : ℚ) /
    (((2 : ℚ) ^ (n - k)) * ((((n - k).factorial : ℕ)) : ℚ)) *
    (((Nat.choose n (2 * k - n) : ℕ)) : ℚ)

Sources

• Schork, On Generalized Stirling Numbers and Normal Ordering, StirW2 formula. Defined independently from the Bessel side.

Used by 1

Used bySecond identification of meromorphic Stirling and Bessel numbers

DefinitionBessel numbers of the first kind#✓fully proved

For natural numbers \(n\) and \(k\) with \(1 \le k \le n\), the Bessel number of the first kind is the rational number

\[b(n,k) = \frac{(-1)^{n-k}\,(2n-k-1)!}{2^{n-k}\,(k-1)!\,(n-k)!}.\]

For other \(n\) and \(k\) the same expression is evaluated with natural-number subtraction truncated at \(0\).

public def besselNumberFirst (n k : ℕ) : ℚ :=
  ((-1 : ℚ) ^ (n - k) * ((((2 * n - k - 1).factorial : ℕ)) : ℚ)) /
    (((2 : ℚ) ^ (n - k)) * ((((k - 1).factorial : ℕ)) : ℚ) *
      ((((n - k).factorial : ℕ)) : ℚ))

Sources

• Schork, On Generalized Stirling Numbers and Normal Ordering, Bessfirst formula. Defined independently from the meromorphic side.

Used by 1

Used byFirst identification of meromorphic Stirling and Bessel numbers

DefinitionBessel numbers of the second kind#✓fully proved

For natural numbers \(n\) and \(k\) with \(k \le n \le 2k\), the Bessel number of the second kind is the rational number

\[B(n,k) = \frac{n!}{2^{n-k}\,(2k-n)!\,(n-k)!}.\]

For other \(n\) and \(k\) the same expression is evaluated with natural-number subtraction truncated at \(0\).

public def besselNumberSecond (n k : ℕ) : ℚ :=
  ((((n.factorial : ℕ))) : ℚ) /
    (((2 : ℚ) ^ (n - k)) * ((((2 * k - n).factorial : ℕ)) : ℚ) *
      ((((n - k).factorial : ℕ)) : ℚ))

Sources

• Schork, On Generalized Stirling Numbers and Normal Ordering, Besssecond formula. Defined independently from the meromorphic side.

Used by 1

Used bySecond identification of meromorphic Stirling and Bessel numbers

TheoremFirst identification of meromorphic Stirling and Bessel numbers#✓fully proved

For all natural numbers \(n\) and \(k\) with \(1 \le k \le n\), the meromorphic Stirling number \(S_{2;-1}(n,k)\) equals the Bessel number of the first kind \(b(n,k)\).

public theorem meromorphicStirlingFirst_eq_besselNumberFirst
    (n k : ℕ) (h1 : 1 ≤ k) (h2 : k ≤ n) :
    meromorphicStirlingFirst n k = besselNumberFirst n k

Uses 2

Statement usesBessel numbers of the first kind · The meromorphic Stirling numbers \(S_{2;-1}(n,k)\)

TheoremSecond identification of meromorphic Stirling and Bessel numbers#✓fully proved

For all natural numbers \(n\) and \(k\) with \(k \le n \le 2k\), the meromorphic Stirling number \(S_{-1;1}(n,k)\) equals the Bessel number of the second kind \(B(n,k)\).

public theorem meromorphicStirlingSecond_eq_besselNumberSecond
    (n k : ℕ) (h1 : n ≤ 2 * k) (h2 : k ≤ n) :
    meromorphicStirlingSecond n k = besselNumberSecond n k

Uses 2

Statement usesBessel numbers of the second kind · The meromorphic Stirling numbers \(S_{-1;1}(n,k)\)

6 fully proved

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