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1: 26.5 Lattice Paths: Catalan Numbers
§26.5 Lattice Paths: Catalan Numbers
§26.5(i) Definitions
C ( n ) is the Catalan number. …
§26.5(ii) Generating Function
§26.5(iii) Recurrence Relations
2: 25.11 Hurwitz Zeta Function
25.11.18 ζ ( 0 , a ) = ln Γ ( a ) - 1 2 ln ( 2 π ) , a > 0 .
25.11.37 k = 1 ( - 1 ) k k ζ ( n k , a ) = - n ln Γ ( a ) + ln ( j = 0 n - 1 Γ ( a - e ( 2 j + 1 ) π i / n ) ) , n = 2 , 3 , 4 , , a 1 .
25.11.39 k = 2 k 2 k ζ ( k + 1 , 3 4 ) = 8 G ,
where G is Catalan’s constant:
25.11.40 G n = 0 ( - 1 ) n ( 2 n + 1 ) 2 = 0.91596 55941 772 .
3: 26.6 Other Lattice Path Numbers
§26.6(iv) Identities
26.6.12 C ( n ) = k = 1 n N ( n , k ) ,
26.6.13 M ( n ) = k = 0 n ( - 1 ) k ( n k ) C ( n + 1 - k ) ,
26.6.14 C ( n ) = k = 0 2 n ( - 1 ) k ( 2 n k ) M ( 2 n - k ) .
4: 26.1 Special Notation
( m n ) binomial coefficient.
C ( n ) Catalan number.
5: 3.12 Mathematical Constants
§3.12 Mathematical Constants
The fundamental constant …Other constants that appear in the DLMF include the base e of natural logarithms …see §4.2(ii), and Euler’s constant γ For access to online high-precision numerical values of mathematical constants see Sloane (2003). …
6: 30.1 Special Notation
x real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, - 1 < x < 1 .
δ arbitrary small positive constant.
The main functions treated in this chapter are the eigenvalues λ n m ( γ 2 ) and the spheroidal wave functions Ps n m ( x , γ 2 ) , Qs n m ( x , γ 2 ) , Ps n m ( z , γ 2 ) , Qs n m ( z , γ 2 ) , and S n m ( j ) ( z , γ ) , j = 1 , 2 , 3 , 4 . …Meixner and Schäfke (1954) use ps , qs , Ps , Qs for Ps , Qs , Ps , Qs , respectively. … Flammer (1957) and Abramowitz and Stegun (1964) use λ m n ( γ ) for λ n m ( γ 2 ) + γ 2 , R m n ( j ) ( γ , z ) for S n m ( j ) ( z , γ ) , and …where d m n ( γ ) is a normalization constant determined by …
7: 16.15 Integral Representations and Integrals
16.15.1 F 1 ( α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( α ) Γ ( γ - α ) 0 1 u α - 1 ( 1 - u ) γ - α - 1 ( 1 - u x ) β ( 1 - u y ) β d u , α > 0 , ( γ - α ) > 0 ,
16.15.2 F 2 ( α ; β , β ; γ , γ ; x , y ) = Γ ( γ ) Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β ) Γ ( γ - β ) 0 1 0 1 u β - 1 v β - 1 ( 1 - u ) γ - β - 1 ( 1 - v ) γ - β - 1 ( 1 - u x - v y ) α d u d v , γ > β > 0 , γ > β > 0 ,
16.15.3 F 3 ( α , α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β - β ) Δ u β - 1 v β - 1 ( 1 - u - v ) γ - β - β - 1 ( 1 - u x ) α ( 1 - v y ) α d u d v , ( γ - β - β ) > 0 , β > 0 , β > 0 ,
16.15.4 F 4 ( α , β ; γ , γ ; x ( 1 - y ) , y ( 1 - x ) ) = Γ ( γ ) Γ ( γ ) Γ ( α ) Γ ( β ) Γ ( γ - α ) Γ ( γ - β ) 0 1 0 1 u α - 1 v β - 1 ( 1 - u ) γ - α - 1 ( 1 - v ) γ - β - 1 ( 1 - u x ) γ + γ - α - 1 ( 1 - v y ) γ + γ - β - 1 ( 1 - u x - v y ) α + β - γ - γ + 1 d u d v , γ > α > 0 , γ > β > 0 .
8: 32.9 Other Elementary Solutions
with κ , λ , μ , and ν arbitrary constants. … with C an arbitrary constant, which is solvable by quadrature. … with κ and μ arbitrary constants. … with C an arbitrary constant, which is solvable by quadrature. … with κ and μ arbitrary constants. …
9: 5.17 Barnes’ G -Function (Double Gamma Function)
G ( z + 1 ) = Γ ( z ) G ( z ) ,
Here B 2 k + 2 is the Bernoulli number (§24.2(i)), and A is Glaisher’s constant, given by
5.17.6 A = e C = 1.28242 71291 00622 63687 ,
5.17.7 C = lim n ( k = 1 n k ln k - ( 1 2 n 2 + 1 2 n + 1 12 ) ln n + 1 4 n 2 ) = γ + ln ( 2 π ) 12 - ζ ( 2 ) 2 π 2 = 1 12 - ζ ( - 1 ) ,
For Glaisher’s constant see also Greene and Knuth (1982, p. 100) and §2.10(i).
10: 30.5 Functions of the Second Kind
Other solutions of (30.2.1) with μ = m , λ = λ n m ( γ 2 ) , and z = x are
30.5.1 Qs n m ( x , γ 2 ) , n = m , m + 1 , m + 2 , .
30.5.2 Qs n m ( - x , γ 2 ) = ( - 1 ) n - m + 1 Qs n m ( x , γ 2 ) ,
30.5.4 𝒲 { Ps n m ( x , γ 2 ) , Qs n m ( x , γ 2 ) } = ( n + m ) ! ( 1 - x 2 ) ( n - m ) ! A n m ( γ 2 ) A n - m ( γ 2 ) ( 0 ) ,
with A n ± m ( γ 2 ) as in (30.11.4). …