id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-1596	Srivastava, H.M.; Vignat, Christophe	Probabilistic Proofs of Some Relationships Between the Bernoulli and Euler Polynomials	2012	11	.pdf	application/pdf	3701	204	78	For n ∈ N0, α ∈ C, a ∈ Cα and any j (1≦ j ≦ α) such that a j 6= 0, B(α)n � x + y|a � = n ∑ k=0 � n k �� B(α)k � y|a � + a j k 2 B(α−1) k−1 � y|a \ a j � � · En−k � x |a j � (32) and E(α)n � x + y|a � = n ∑ k=0 2 k+ 1 � 1 a j E(α−1) k+1 � y|a \ a j � − 1 a j E(α)k+1 � y|a � � · B(1)n−k � x |a j � , (33) where a \ a j := � a1, · · · , a j−1, a j+1, · · · , aα � . Starting from the identity (5) with α = 1, if we replace x and y by x a j and y a j , respectively, we obtain E �� x a j + y a j + � ıL ( j) B − 1 2 � �n� = E � n ∑ k=0 � n k � · �� y a j + � ıL ( j) B − 1 2 � �k + k 2 � y a j �k−1�� x a j + ıLE − 1 2 �n−k� , which, when multiplied by an j on both sides, yields E �� x + y + a j � ıL ( j) B − 1 2 � �n� = E � n ∑ k=0 � n k � · �� y + a j � ıL ( j) B − 1 2 � �k + k 2 a j yk−1 �� x + a j � ıLE − 1 2 � �n−k� .	cache/ejpam-1596.pdf	txt/ejpam-1596.txt