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Higher-order non-holomorphic Eisenstein series associated to a Fuchsian group  are defined by twisting the series expansion for classical non-holomorphic Eisenstein series by powers of modular symbols. Their functional identities include multiplicative and additive factors, making them distinct from classical Eisenstein series. In this article we prove the meromorphic continuation of these series and establish their functional equations which relate values at  and . In addition, we construct...
    			                    
    			                 
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    			                    
                                       
We consider a class of linear ordinary differential equations determined by a modular form of weight one, and construct vector-valued modular forms of weight two by using solutions of such differential equations.
    			                    
    			                 
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    			
    			 
 
    			
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