JetssheetaDumcb

JetssheetaDumcb

Answered

2022-10-22

Prove that k = 0 m ( m k ) ( n k ) = n + 1 n + 1 m

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Answer & Explanation

Kason Gonzales

Kason Gonzales

Expert

2022-10-23Added 15 answers

Step 1
You can solve your problem by following these steps:
1) First show that
( m k ) ( n k ) = ( n k m k ) ( n m ) .
2) Now show that
( n m ) n + 1 n + 1 m = ( n + 1 m ) .
3) Use steps 1 and 2 to prove that the equation you require is equivalent to proving
k = 0 m ( n k m k ) = ( n + 1 m ) .
4) Now prove the relation:
k = 0 m ( n k m k ) = ( n + 1 m ) .
Step 2
4a) You can prove this relation in various ways. One of them is to write ( n m 0 ) as ( n m + 1 0 ) and then repeatedly use the relation ( x r ) + ( x r 1 ) = ( x + 1 r ) . Another method is induction.

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mafalexpicsak

mafalexpicsak

Expert

2022-10-24Added 4 answers

Step 1
k = 0 m ( m k ) ( n k ) = n + 1 n + 1 m :   ?
Step 2
k = 0 m ( m k ) ( n k ) = ( n + 1 ) k = 0 m ( m k ) Γ ( k + 1 ) Γ ( n k + 1 ) Γ ( n + 2 ) =   ( n + 1 ) k = 0 m ( m k ) 0 1 t k ( 1 t ) n k d t =   ( n + 1 ) 0 1 ( 1 t ) n k = 0 m ( m k ) ( t 1 t ) k d t =   ( n + 1 ) 0 1 ( 1 t ) n ( 1 + t 1 t ) m d t =   ( n + 1 ) 0 1 ( 1 t ) n m d t = n + 1 n + 1 m

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