The linear differential operators U γ c → γ c + 1 +, L σ c → σ c − 1 − are called the step-up and step-down operators for the upper and lower indices, respectively. Similar relations also exist for the lower parameters: H ( γ → σ → − e → c x → ) = 1 σ c − 1 ( ∑ b = 1 r ν c b x b ∂ ∂ x b + σ c − 1 ) H ( γ → σ → x → ) ≡ L − H ( γ → σ → x → ). Two functions of type (1) with sets of parameters shifted by unity, H ( γ → + e c → σ → x → ) and H ( γ → σ → x → ), are related by a linear differential operator: H ( γ → + e c → σ → x → ) = 1 γ c ( ∑ a = 1 r μ c a x a ∂ ∂ x a + γ c ) H ( γ → σ → x → ) ≡ U + H ( γ →, σ →, x → ).

Let e → j = ( 0, …, 0, 1, 0, …, 0 ) denote the unit vector with unity in its jth entry, and let us define x → m → = x 1 m 1 ⋯ x r m r for any integer multi-index m → = ( m 1, …, m r ).

The sequences γ → = ( γ 1, …, γ K ) and σ → = ( σ 1, …, σ L ) are called upper and lower parameters of the hypergeometric function, respectively. Let us consider a multiple series: H ( γ → σ → x → ) = ∑ m 1, m 2, …, m r = 0 ∞ ( Π j = 1 K Γ ( ∑ a = 1 r μ j a m a + γ j ) Γ − 1 ( γ j ) Π k = 1 L Γ ( ∑ b = 1 r ν k b m b + σ k ) Γ − 1 ( σ k ) ) x 1 m 1 ⋯ x r m r, with μ a b, ν a b ∈ Z, γ j, σ k ∈ C. In particular, a large class of Feynman diagrams is expressed in terms of Horn-type hypergeometric functions. Multiple hypergeometric functions, ,, play an important role in many branches of science.

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